# lfunc_search downloaded from the LMFDB on 04 July 2026. # Search link: https://www.lmfdb.org/L/2/2667 # Query "{'degree': 2, 'conductor': 2667}" returned 345 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. "2-2667-2667.101-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.101" [] [[0.0, 0.0]] 0 true true false false -0.3587241232329261 0 0.60171763854094182185531339652 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/101/1"] "2-2667-2667.1010-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1010" [] [[0.0, 0.0]] 0 true true false false 0.1720169102239171 0 1.55121692120778891524627540939 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1010/1"] "2-2667-2667.1031-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1031" [] [[0.0, 0.0]] 0 true true false false -0.03995570626686678 0 1.49988185308750994446547536088 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1031/1"] "2-2667-2667.1049-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1049" [] [[0.0, 0.0]] 0 true true false false -0.1752988181714087 0 0.68467871254745100482746954610 ["ModularForm/GL2/Q/holomorphic/2667/1/dy/b/1049/1"] "2-2667-2667.1049-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1049" [] [[0.0, 0.0]] 0 true true false false -0.032441675314265846 0 1.19775598902225158805937307409 ["ModularForm/GL2/Q/holomorphic/2667/1/dy/a/1049/1"] "2-2667-2667.1052-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1052" [] [[0.0, 0.0]] 0 true true false false 0.13004059584230884 0 1.45440941061742572211464644870 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1052/1"] "2-2667-2667.1055-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1055" [] [[0.0, 0.0]] 0 true true false false 0.3353018004440104 0 2.20212976812988247447008573068 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1055/1"] "2-2667-2667.107-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.107" [] [[0.0, 0.0]] 0 true true false false 0.09867007541371052 0 1.07667477996975878095225522386 ["ModularForm/GL2/Q/holomorphic/2667/1/bf/a/107/1"] "2-2667-2667.107-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.107" [] [[0.0, 0.0]] 0 true true false false 0.09867007541371052 0 1.37720298372323427136629263249 ["ModularForm/GL2/Q/holomorphic/2667/1/bf/a/107/2"] "2-2667-2667.1070-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1070" [] [[0.0, 0.0]] 0 true true false false 0.3675202035141877 0 1.62078625653319215004182770886 ["ModularForm/GL2/Q/holomorphic/2667/1/dy/a/1070/1"] "2-2667-2667.1070-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1070" [] [[0.0, 0.0]] 0 true true false false 0.08180591779990198 0 1.75623126789353345235107100646 ["ModularForm/GL2/Q/holomorphic/2667/1/dy/b/1070/1"] "2-2667-2667.11-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.11" [] [[0.0, 0.0]] 0 true true false false -0.25610658734520264 0 0.50739429815484118815325650830 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/11/1"] "2-2667-2667.110-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.110" [] [[0.0, 0.0]] 0 true true false false 0.22658314306202165 0 1.18963419225398483535035233250 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/110/1"] "2-2667-2667.1109-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1109" [] [[0.0, 0.0]] 0 true true false false -0.12878796974539153 0 0.70533032576244653229769946323 ["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1109/1"] "2-2667-2667.1115-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1115" [] [[0.0, 0.0]] 0 true true false false 0.1901898428023464 0 1.20293841686557066281040141728 ["ModularForm/GL2/Q/holomorphic/2667/1/cj/a/1115/1"] "2-2667-2667.1130-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1130" [] [[0.0, 0.0]] 0 true true false false -0.3353018004440104 0 0.892531483649218376880896555421 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1130/1"] "2-2667-2667.1136-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1136" [] [[0.0, 0.0]] 0 true true false false 0.022561039793695455 0 1.09831728403555716247009293283 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1136/1"] "2-2667-2667.1172-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1172" [] [[0.0, 0.0]] 0 true true false false -0.05360409781166061 0 0.78137105624400628338287704326 ["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1172/1"] "2-2667-2667.1178-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1178" [] [[0.0, 0.0]] 0 true true false false 0.035716752744805715 0 1.61637543024348669977713569047 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1178/1"] "2-2667-2667.1187-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1187" [] [[0.0, 0.0]] 0 true true false false 0.4324805326719127 0 0.22554051342086438328318076123 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1187/1"] "2-2667-2667.1202-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1202" [] [[0.0, 0.0]] 0 true true false false 0.4218327822774399 0 1.98814587211582674147134741206 ["ModularForm/GL2/Q/holomorphic/2667/1/cn/a/1202/1"] "2-2667-2667.1235-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1235" [] [[0.0, 0.0]] 0 true true false false 0.20373438217995887 0 1.15101658537747722884770721666 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1235/1"] "2-2667-2667.1238-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1238" [] [[0.0, 0.0]] 0 true true false false -0.24370345208373562 0 0.31272497640004543566384253630 ["ModularForm/GL2/Q/holomorphic/2667/1/by/a/1238/1"] "2-2667-2667.1238-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1238" [] [[0.0, 0.0]] 0 true true false false 0.32772511934483584 0 1.93296449299934845058115657757 ["ModularForm/GL2/Q/holomorphic/2667/1/by/b/1238/1"] "2-2667-2667.1241-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1241" [] [[0.0, 0.0]] 0 true true false false -0.41303256294160895 0 0.73984357784957977153826997070 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1241/1"] "2-2667-2667.1244-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1244" [] [[0.0, 0.0]] 0 true true false false -0.36599726474408145 0 0.30306877153124069410260429240 ["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1244/1"] "2-2667-2667.125-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.125" [] [[0.0, 0.0]] 0 true true false false -0.4125589865581766 0 0.65668807559335159002550500562 ["ModularForm/GL2/Q/holomorphic/2667/1/by/a/125/1"] "2-2667-2667.125-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.125" [] [[0.0, 0.0]] 0 true true false false 0.3017267277275377 0 1.88698893373596590562006438384 ["ModularForm/GL2/Q/holomorphic/2667/1/by/b/125/1"] "2-2667-2667.1250-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1250" [] [[0.0, 0.0]] 0 true true false false -0.04782363819901003 0 1.19346832478074838323665877949 ["ModularForm/GL2/Q/holomorphic/2667/1/bi/a/1250/2", "ArtinRepresentation/2.2667.16t60.a.d"] "2-2667-2667.1250-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1250" [] [[0.0, 0.0]] 0 true true false false -0.04782363819901003 0 1.58970722134816728131174151537 ["ModularForm/GL2/Q/holomorphic/2667/1/bi/a/1250/1", "ArtinRepresentation/2.2667.16t60.a.c"] "2-2667-2667.1277-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1277" [] [[0.0, 0.0]] 0 true true false false -0.05184413568435023 0 0.898849933768470370463972887188 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1277/1"] "2-2667-2667.1280-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1280" [] [[0.0, 0.0]] 0 true true false false -0.039795084169351895 0 1.15595441705727166532278464102 ["ModularForm/GL2/Q/holomorphic/2667/1/dy/b/1280/1"] "2-2667-2667.1280-c0-0-1" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1280" [] [[0.0, 0.0]] 0 true true false false 0.24591920154493382 0 2.08427281121610655966481328094 ["ModularForm/GL2/Q/holomorphic/2667/1/dy/a/1280/1"] "2-2667-2667.1283-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1283" [] [[0.0, 0.0]] 0 true true false false -0.11567011251330785 0 0.816332514993940254193822654057 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1283/1"] "2-2667-2667.1292-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1292" [] [[0.0, 0.0]] 0 true true false false 0.14203629996613207 0 1.36700045120958121851046359562 ["ModularForm/GL2/Q/holomorphic/2667/1/cj/a/1292/1"] "2-2667-2667.1298-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1298" [] [[0.0, 0.0]] 0 true true false false -0.4218327822774399 0 0.50720949699666962138672927268 ["ModularForm/GL2/Q/holomorphic/2667/1/cn/a/1298/1"] "2-2667-2667.1304-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1304" [] [[0.0, 0.0]] 0 true true false false -0.41181700058376725 0 1.79203629098479472180969427913 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1304/1"] "2-2667-2667.1328-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1328" [] [[0.0, 0.0]] 0 true true false false 0.266724274463395 0 1.19633186681374278428653025434 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1328/1"] "2-2667-2667.1361-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1361" [] [[0.0, 0.0]] 0 true true false false 0.2982582993842134 0 1.87568067307921799879435674238 ["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1361/1"] "2-2667-2667.1382-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1382" [] [[0.0, 0.0]] 0 true true false false -0.22658314306202165 0 0.03499118923626855400275214086 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1382/1"] "2-2667-2667.1403-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1403" [] [[0.0, 0.0]] 0 true true false false 0.10451739923111228 0 1.68572432552879886900597161504 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1403/1"] "2-2667-2667.1418-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1418" [] [[0.0, 0.0]] 0 true true false false -0.1720169102239171 0 0.36223513579917873056480391643 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1418/1"] "2-2667-2667.1439-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1439" [] [[0.0, 0.0]] 0 true 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["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1454/1"] "2-2667-2667.1475-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1475" [] [[0.0, 0.0]] 0 true true false false -0.08968167600554684 0 1.49437874672220028709620626728 ["ModularForm/GL2/Q/holomorphic/2667/1/ei/a/1475/1"] "2-2667-2667.1481-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1481" [] [[0.0, 0.0]] 0 true true false false 0.1575818856256457 0 1.52148885385653211517621177601 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1481/1"] "2-2667-2667.1487-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1487" [] [[0.0, 0.0]] 0 true true false false 0.16968698511119598 0 1.26503888835074937181999871223 ["ModularForm/GL2/Q/holomorphic/2667/1/ck/a/1487/1"] "2-2667-2667.149-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.149" [] [[0.0, 0.0]] 0 true true false false -0.3781366530878141 0 0.44724756586763841191253940685 ["ModularForm/GL2/Q/holomorphic/2667/1/co/a/149/1"] "2-2667-2667.1535-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1535" [] [[0.0, 0.0]] 0 true true false false -0.061993826113434465 0 0.926520903345146256009677159970 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1535/1"] "2-2667-2667.1538-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1538" [] [[0.0, 0.0]] 0 true true false false 0.1333706543904506 0 0.944605037146697896216194148607 ["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1538/1"] "2-2667-2667.1565-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1565" [] [[0.0, 0.0]] 0 true true false false 0.1539031549649583 0 1.28032338912707101272350001098 ["ModularForm/GL2/Q/holomorphic/2667/1/em/a/1565/1"] "2-2667-2667.158-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.158" [] [[0.0, 0.0]] 0 true true false false -0.06230163048244073 0 0.950347785176427753871301104626 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/158/1"] "2-2667-2667.1580-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1580" [] [[0.0, 0.0]] 0 true true false false 0.12878796974539153 0 1.45756843057551635198869128467 ["ModularForm/GL2/Q/holomorphic/2667/1/en/a/1580/1"] "2-2667-2667.1586-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1586" [] [[0.0, 0.0]] 0 true true false false 0.21973392457115756 0 1.39425845808011055884405954099 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1586/1"] "2-2667-2667.1598-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1598" [] [[0.0, 0.0]] 0 true true false false 0.45593467598511267 0 2.35670768292006930810395121986 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1598/1"] "2-2667-2667.1628-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1628" [] [[0.0, 0.0]] 0 true true false false 0.061993826113434465 0 1.37581101292237305451415329482 ["ModularForm/GL2/Q/holomorphic/2667/1/ej/a/1628/1"] "2-2667-2667.1634-c0-0-0" 1.1536925017924913 1.3310063886922174 2 2667 "2667.1634" [] [[0.0, 0.0]] 0 true true false false 0.09099914171603417 0 1.47774596903819095642852626290 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"ModularForm/GL2/Q/holomorphic/2667/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/2667/2/a/c"] "2-2667-1.1-c1-0-98" 4.614770007169966 21.29610221907549 2 2667 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.52737954271804143232303951405 ["ModularForm/GL2/Q/holomorphic/2667/2/a/p/1/14"] "2-2667-1.1-c1-0-99" 4.614770007169966 21.29610221907549 2 2667 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.53251657745741075523001195742 ["ModularForm/GL2/Q/holomorphic/2667/2/a/p/1/11"] # 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).