# lfunc_search downloaded from the LMFDB on 03 July 2026. # Search link: https://www.lmfdb.org/L/2/2904/264.125/c0-0 # Query "{'degree': 2, 'conductor': 2904, 'spectral_label': 'c0-0'}" returned 292 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "2-2904-24.5-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.55187911663421358830678833432 ["ModularForm/GL2/Q/holomorphic/2904/1/n/a/485/1"] "2-2904-24.5-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 0.893931504983716376623420509794 ["ModularForm/GL2/Q/holomorphic/2904/1/n/b/485/1"] "2-2904-24.5-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.14881480700443904393048164395 ["ModularForm/GL2/Q/holomorphic/2904/1/n/a/485/2"] "2-2904-24.5-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.30427514213008840625615469403 ["ModularForm/GL2/Q/holomorphic/2904/1/n/c/485/1"] "2-2904-24.5-c0-0-4" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.44168443712433847279911511715 ["ModularForm/GL2/Q/holomorphic/2904/1/n/c/485/2"] "2-2904-24.5-c0-0-5" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.51105156472123275476572473348 ["ModularForm/GL2/Q/holomorphic/2904/1/n/d/485/1", "ArtinRepresentation/2.2904.5t2.a.b"] "2-2904-24.5-c0-0-6" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.58760603174498514268413740911 ["ModularForm/GL2/Q/holomorphic/2904/1/n/b/485/2"] "2-2904-24.5-c0-0-7" 1.2038624492383834 1.449284796686239 2 2904 "24.5" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.69450777453644932374480052484 ["ModularForm/GL2/Q/holomorphic/2904/1/n/d/485/2", "ArtinRepresentation/2.2904.5t2.a.a"] "2-2904-264.107-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.3124315125347866 0 0.19135453009396812545294954508 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/1691/3"] "2-2904-264.107-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.0410676202934349 0 0.837565304214172443143296394992 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/1691/1"] "2-2904-264.107-c0-0-10" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.3037837371021717 0 2.00776542836197641795962796891 ["ModularForm/GL2/Q/holomorphic/2904/1/r/g/1691/1"] "2-2904-264.107-c0-0-11" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false -0.4962162628978283 0 2.10906987402292325163792147184 ["ModularForm/GL2/Q/holomorphic/2904/1/r/f/1691/1"] "2-2904-264.107-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.0037837371021717274 0 0.865678770490946496922066828240 ["ModularForm/GL2/Q/holomorphic/2904/1/r/b/1691/1"] "2-2904-264.107-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false -0.18756848746521343 0 0.883184408390101345609889964756 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/1691/2"] "2-2904-264.107-c0-0-4" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false -0.19621626289782826 0 0.939310772716231681941229867000 ["ModularForm/GL2/Q/holomorphic/2904/1/r/d/1691/1"] "2-2904-264.107-c0-0-5" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false -0.4589323797065651 0 1.16049365443521932595854220695 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/1691/4"] "2-2904-264.107-c0-0-6" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.24971539572604978 0 1.36648481880990558515463395386 ["ModularForm/GL2/Q/holomorphic/2904/1/r/e/1691/1"] "2-2904-264.107-c0-0-7" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.40378373710217175 0 1.52600127436587035706300352040 ["ModularForm/GL2/Q/holomorphic/2904/1/r/a/1691/1"] "2-2904-264.107-c0-0-8" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.14971539572604978 0 1.53351391506299352948068221513 ["ModularForm/GL2/Q/holomorphic/2904/1/r/c/1691/1"] "2-2904-264.107-c0-0-9" 1.2038624492383834 1.449284796686239 2 2904 "264.107" [] [[0.0, 0.0]] 0 true true false false 0.10378373710217174 0 1.60004084426365855893424139988 ["ModularForm/GL2/Q/holomorphic/2904/1/r/h/1691/1"] "2-2904-264.125-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "264.125" [] [[0.0, 0.0]] 0 true true false false 0.361066607032565 0 0.53757576617177220894676207616 ["ModularForm/GL2/Q/holomorphic/2904/1/t/f/2501/1"] "2-2904-264.125-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "264.125" [] [[0.0, 0.0]] 0 true true false false -0.23893339296743507 0 0.70907795786980880606332349250 ["ModularForm/GL2/Q/holomorphic/2904/1/t/c/2501/1"] "2-2904-264.125-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "264.125" [] [[0.0, 0.0]] 0 true true false false 0.08543426013921357 0 0.892659245867887702410763283673 ["ModularForm/GL2/Q/holomorphic/2904/1/t/b/2501/1"] "2-2904-264.125-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "264.125" [] [[0.0, 0.0]] 0 true true false false -0.23893339296743507 0 0.906541681093607830601287172479 ["ModularForm/GL2/Q/holomorphic/2904/1/t/d/2501/1"] "2-2904-264.125-c0-0-4" 1.2038624492383834 1.449284796686239 2 2904 "264.125" [] [[0.0, 0.0]] 0 true true false false 0.361066607032565 0 1.62986697657994633061680917106 ["ModularForm/GL2/Q/holomorphic/2904/1/t/a/2501/1"] "2-2904-264.125-c0-0-5" 1.2038624492383834 1.449284796686239 2 2904 "264.125" [] [[0.0, 0.0]] 0 true true false false 0.08543426013921357 0 1.90914899624847906112867625086 ["ModularForm/GL2/Q/holomorphic/2904/1/t/e/2501/1"] "2-2904-264.131-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false -0.2648503441347367 0 0.19608969037920413481043053739 ["ModularForm/GL2/Q/holomorphic/2904/1/p/a/1451/1"] "2-2904-264.131-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false -0.32010944950188586 0 0.76053187841923595533972068810 ["ModularForm/GL2/Q/holomorphic/2904/1/p/b/1451/3"] "2-2904-264.131-c0-0-10" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false -0.08921799724138531 0 1.60893736557813558853162853943 ["ModularForm/GL2/Q/holomorphic/2904/1/p/c/1451/4"] "2-2904-264.131-c0-0-11" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false 0.08921799724138531 0 1.80842415462869066372663511175 ["ModularForm/GL2/Q/holomorphic/2904/1/p/c/1451/3"] "2-2904-264.131-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false -0.21078200275861472 0 0.850287365950519648052908323138 ["ModularForm/GL2/Q/holomorphic/2904/1/p/a/1451/4"] "2-2904-264.131-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false -0.1648503441347367 0 1.17566368047956827955994987835 ["ModularForm/GL2/Q/holomorphic/2904/1/p/c/1451/2"] "2-2904-264.131-c0-0-4" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false 0.32010944950188586 0 1.33737969620750642756655662545 ["ModularForm/GL2/Q/holomorphic/2904/1/p/b/1451/1"] "2-2904-264.131-c0-0-5" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false 0.2648503441347367 0 1.43541158413463600243659622644 ["ModularForm/GL2/Q/holomorphic/2904/1/p/a/1451/2"] "2-2904-264.131-c0-0-6" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false 0.17989055049811414 0 1.47545078631259128152162531697 ["ModularForm/GL2/Q/holomorphic/2904/1/p/b/1451/2"] "2-2904-264.131-c0-0-7" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false 0.21078200275861472 0 1.49716721724941086382056009714 ["ModularForm/GL2/Q/holomorphic/2904/1/p/a/1451/3"] "2-2904-264.131-c0-0-8" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false -0.17989055049811414 0 1.51933748378733798441512312136 ["ModularForm/GL2/Q/holomorphic/2904/1/p/b/1451/4"] "2-2904-264.131-c0-0-9" 1.2038624492383834 1.449284796686239 2 2904 "264.131" [] [[0.0, 0.0]] 0 true true false false 0.1648503441347367 0 1.55838096717932256640498570232 ["ModularForm/GL2/Q/holomorphic/2904/1/p/c/1451/1"] "2-2904-264.227-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.40378373710217175 0 0.04316559917529121968754823266 ["ModularForm/GL2/Q/holomorphic/2904/1/r/a/2339/1"] "2-2904-264.227-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false 0.4962162628978283 0 0.62743601158312917463348851989 ["ModularForm/GL2/Q/holomorphic/2904/1/r/f/2339/1"] "2-2904-264.227-c0-0-10" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false 0.4589323797065651 0 2.22673352946006381534607482119 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/2339/4"] "2-2904-264.227-c0-0-11" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.3124315125347866 0 2.34284061798900140433577315986 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/2339/3"] "2-2904-264.227-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.14971539572604978 0 0.795780808654690706120819983921 ["ModularForm/GL2/Q/holomorphic/2904/1/r/c/2339/1"] "2-2904-264.227-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.0410676202934349 0 0.852221120218520765805551593833 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/2339/1"] "2-2904-264.227-c0-0-4" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.24971539572604978 0 0.925021892561711100216839836474 ["ModularForm/GL2/Q/holomorphic/2904/1/r/e/2339/1"] "2-2904-264.227-c0-0-5" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false 0.19621626289782826 0 0.925984673318774566175082084492 ["ModularForm/GL2/Q/holomorphic/2904/1/r/d/2339/1"] "2-2904-264.227-c0-0-6" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.0037837371021717274 0 0.977784251119320904949550260392 ["ModularForm/GL2/Q/holomorphic/2904/1/r/b/2339/1"] "2-2904-264.227-c0-0-7" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.3037837371021717 0 1.22334024299047256513159098165 ["ModularForm/GL2/Q/holomorphic/2904/1/r/g/2339/1"] "2-2904-264.227-c0-0-8" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false -0.10378373710217174 0 1.38636028608583016023443986518 ["ModularForm/GL2/Q/holomorphic/2904/1/r/h/2339/1"] "2-2904-264.227-c0-0-9" 1.2038624492383834 1.449284796686239 2 2904 "264.227" [] [[0.0, 0.0]] 0 true true false false 0.18756848746521343 0 1.63873857338055313798199239080 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/2339/2"] "2-2904-264.245-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "264.245" [] [[0.0, 0.0]] 0 true true false false -0.08543426013921357 0 0.44499516561161897979251559974 ["ModularForm/GL2/Q/holomorphic/2904/1/t/b/245/1"] "2-2904-264.245-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "264.245" [] [[0.0, 0.0]] 0 true true false false -0.361066607032565 0 0.910882486956105059015886570375 ["ModularForm/GL2/Q/holomorphic/2904/1/t/a/245/1"] "2-2904-264.245-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "264.245" [] [[0.0, 0.0]] 0 true true false false -0.08543426013921357 0 1.29765306243534138113142285422 ["ModularForm/GL2/Q/holomorphic/2904/1/t/e/245/1"] "2-2904-264.245-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "264.245" [] [[0.0, 0.0]] 0 true true false false 0.23893339296743507 0 1.46284806517909937039947640391 ["ModularForm/GL2/Q/holomorphic/2904/1/t/c/245/1"] "2-2904-264.245-c0-0-4" 1.2038624492383834 1.449284796686239 2 2904 "264.245" [] [[0.0, 0.0]] 0 true true false false 0.23893339296743507 0 1.86230757303474009071058746396 ["ModularForm/GL2/Q/holomorphic/2904/1/t/d/245/1"] "2-2904-264.245-c0-0-5" 1.2038624492383834 1.449284796686239 2 2904 "264.245" [] [[0.0, 0.0]] 0 true true false false -0.361066607032565 0 2.55490483853617166275369334005 ["ModularForm/GL2/Q/holomorphic/2904/1/t/f/245/1"] "2-2904-264.35-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "264.35" [] [[0.0, 0.0]] 0 true true false false -0.48503919310250754 0 0.25932039952091991196125630776 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/2411/2"] "2-2904-264.35-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "264.35" [] [[0.0, 0.0]] 0 true true false false -0.45728286993039324 0 0.63061550161449160570920647751 ["ModularForm/GL2/Q/holomorphic/2904/1/r/g/2411/1"] "2-2904-264.35-c0-0-10" 1.2038624492383834 1.449284796686239 2 2904 "264.35" [] [[0.0, 0.0]] 0 true true false false 0.3183494769629582 0 1.39015566969009914972765690126 ["ModularForm/GL2/Q/holomorphic/2904/1/r/c/2411/1"] "2-2904-264.35-c0-0-11" 1.2038624492383834 1.449284796686239 2 2904 "264.35" [] [[0.0, 0.0]] 0 true true false false 0.4427171300696068 0 1.88229607111486082395928508580 ["ModularForm/GL2/Q/holomorphic/2904/1/r/b/2411/1"] "2-2904-264.35-c0-0-2" 1.2038624492383834 1.449284796686239 2 2904 "264.35" [] [[0.0, 0.0]] 0 true true false false 0.14610580013507246 0 0.957877416583690577001028276566 ["ModularForm/GL2/Q/holomorphic/2904/1/r/i/2411/1"] "2-2904-264.35-c0-0-3" 1.2038624492383834 1.449284796686239 2 2904 "264.35" [] [[0.0, 0.0]] 0 true true false false 0.2427171300696068 0 0.987407674130728750104256047466 ["ModularForm/GL2/Q/holomorphic/2904/1/r/a/2411/1"] "2-2904-264.35-c0-0-4" 1.2038624492383834 1.449284796686239 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"2-2904-2904.917-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "2904.917" [] [[0.0, 0.0]] 0 true true false false 0.4991735537190083 0 2.40499266096053074164723246390 ["ModularForm/GL2/Q/holomorphic/2904/1/cj/b/917/1"] "2-2904-2904.923-c0-0-0" 1.2038624492383834 1.449284796686239 2 2904 "2904.923" [] [[0.0, 0.0]] 0 true true false false -0.47314049586776863 0 0.897919799558461194219611331851 ["ModularForm/GL2/Q/holomorphic/2904/1/bh/b/923/1"] "2-2904-2904.923-c0-0-1" 1.2038624492383834 1.449284796686239 2 2904 "2904.923" [] [[0.0, 0.0]] 0 true true false false -0.42768595041322316 0 1.73264509694488884828827804237 ["ModularForm/GL2/Q/holomorphic/2904/1/bh/a/923/1"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).