# lfunc_search downloaded from the LMFDB on 21 May 2026. # Search link: https://www.lmfdb.org/L/2/6^2 # Query "{'degree': 2, 'conductor': 36}" returned 3177 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. "2-6e2-1.1-c1-0-0" 0.5361543129329555 0.2874614472766095 2 36 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 3.44334336790947687892993758586 ["EllipticCurve/Q/36/a", "ModularForm/GL2/Q/holomorphic/36/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/36/2/a/a"] "2-6e2-12.11-c1-0-0" 0.5361543129329555 0.2874614472766095 2 36 "12.11" [] [[0.5, 0.0]] 1 true true false false -0.1520433619923482 0 2.88466240953240789052920623905 ["ModularForm/GL2/Q/holomorphic/36/2/b/a/35/2"] "2-6e2-12.11-c1-0-1" 0.5361543129329555 0.2874614472766095 2 36 "12.11" [] [[0.5, 0.0]] 1 true true false false 0.1520433619923482 0 4.43545401389281113981475638487 ["ModularForm/GL2/Q/holomorphic/36/2/b/a/35/1"] "2-6e2-36.11-c1-0-0" 0.5361543129329555 0.2874614472766095 2 36 "36.11" [] [[0.5, 0.0]] 1 true true false false -0.07457264680669481 0 2.44171528296503042666414328994 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/11/1"] "2-6e2-36.11-c1-0-1" 0.5361543129329555 0.2874614472766095 2 36 "36.11" [] [[0.5, 0.0]] 1 true true false false -0.10920244842592149 0 3.21460605818579881970173812500 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/11/3"] "2-6e2-36.11-c1-0-2" 0.5361543129329555 0.2874614472766095 2 36 "36.11" [] [[0.5, 0.0]] 1 true true false false 0.0715432546428084 0 4.28240964674254658949224646414 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/11/4"] "2-6e2-36.11-c1-0-3" 0.5361543129329555 0.2874614472766095 2 36 "36.11" [] [[0.5, 0.0]] 1 true true false false 0.22334295170091906 0 4.62632618248830518501790350783 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/11/2"] "2-6e2-36.23-c1-0-0" 0.5361543129329555 0.2874614472766095 2 36 "36.23" [] [[0.5, 0.0]] 1 true true false false -0.22334295170091906 0 2.12226860007967066366062149714 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/23/2"] "2-6e2-36.23-c1-0-1" 0.5361543129329555 0.2874614472766095 2 36 "36.23" [] [[0.5, 0.0]] 1 true true false false 0.07457264680669481 0 3.35888230345847000471252667227 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/23/1"] "2-6e2-36.23-c1-0-2" 0.5361543129329555 0.2874614472766095 2 36 "36.23" [] [[0.5, 0.0]] 1 true true false false -0.0715432546428084 0 3.66052421143855954487685300223 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/23/4"] "2-6e2-36.23-c1-0-3" 0.5361543129329555 0.2874614472766095 2 36 "36.23" [] [[0.5, 0.0]] 1 true true false false 0.10920244842592149 0 3.85338976274152635267506907552 ["ModularForm/GL2/Q/holomorphic/36/2/h/a/23/3"] "2-6e2-9.4-c1-0-0" 0.5361543129329555 0.2874614472766095 2 36 "9.4" [] [[0.5, 0.0]] 1 true true false false 0.05555555555555555 0 4.04092755146920523894353866314 ["ModularForm/GL2/Q/holomorphic/36/2/e/a/13/1"] "2-6e2-9.7-c1-0-0" 0.5361543129329555 0.2874614472766095 2 36 "9.7" [] [[0.5, 0.0]] 1 true true false false -0.05555555555555555 0 3.03859835267122871133236527477 ["ModularForm/GL2/Q/holomorphic/36/2/e/a/25/1"] "2-6e2-36.31-c2-0-0" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false -0.32211840199332165 0 0.65066533261453962682350270084 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/1"] "2-6e2-36.31-c2-0-1" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false -0.2777777777777778 0 1.70470058177875413975395387501 ["ModularForm/GL2/Q/holomorphic/36/3/f/a/31/1"] "2-6e2-36.31-c2-0-2" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false -0.21364689317643626 0 2.11262144784266889695372323625 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/7"] "2-6e2-36.31-c2-0-3" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false 0.06886752507952705 0 2.55650648891392223678608191777 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/3"] "2-6e2-36.31-c2-0-4" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false 0.12284145812596474 0 2.92130820260622556641288508592 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/2"] "2-6e2-36.31-c2-0-5" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false -0.11159659284141832 0 2.93214743671620789930773005350 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/6"] "2-6e2-36.31-c2-0-6" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false 0.04398499199838281 0 3.21588844703711270142071858410 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/8"] "2-6e2-36.31-c2-0-7" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false 0.05555555555555555 0 3.75814638097851894193692459702 ["ModularForm/GL2/Q/holomorphic/36/3/f/b/31/1"] "2-6e2-36.31-c2-0-8" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false 0.17995453523132765 0 4.02489187222017173324061173990 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/5"] "2-6e2-36.31-c2-0-9" 0.9904185739863065 0.980928951697069 2 36 "36.31" [] [[1.0, 0.0]] 2 true true false false 0.34282448868708515 0 4.44257723734643790588761038869 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/31/4"] "2-6e2-36.7-c2-0-0" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false -0.34282448868708515 0 1.34574200029128207473128367631 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/4"] "2-6e2-36.7-c2-0-1" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false -0.12284145812596474 0 1.60467221093575832094185885641 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/2"] "2-6e2-36.7-c2-0-2" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false -0.06886752507952705 0 2.35306030444287242496244825486 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/3"] "2-6e2-36.7-c2-0-3" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false -0.17995453523132765 0 2.59811354296872973494939135639 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/5"] "2-6e2-36.7-c2-0-4" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false -0.05555555555555555 0 2.89271783766293095826803549691 ["ModularForm/GL2/Q/holomorphic/36/3/f/b/7/1"] "2-6e2-36.7-c2-0-5" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false -0.04398499199838281 0 3.19808680226487605547216138440 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/8"] "2-6e2-36.7-c2-0-6" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false 0.11159659284141832 0 3.39813769776261327187346693904 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/6"] "2-6e2-36.7-c2-0-7" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false 0.32211840199332165 0 3.44418080994491580461684732215 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/1"] "2-6e2-36.7-c2-0-8" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false 0.2777777777777778 0 3.91804359982382657055054831934 ["ModularForm/GL2/Q/holomorphic/36/3/f/a/7/1"] "2-6e2-36.7-c2-0-9" 0.9904185739863065 0.980928951697069 2 36 "36.7" [] [[1.0, 0.0]] 2 true true false false 0.21364689317643626 0 4.43831727439837761211662327228 ["ModularForm/GL2/Q/holomorphic/36/3/f/c/7/7"] "2-6e2-4.3-c2-0-0" 0.9904185739863065 0.980928951697069 2 36 "4.3" [] [[1.0, 0.0]] 2 true true true true 0.0 0 2.18399671879391988933381425506 ["ModularForm/GL2/Q/holomorphic/36/3/d/a/19/1", "ModularForm/GL2/Q/holomorphic/36/3/d/a"] "2-6e2-4.3-c2-0-1" 0.9904185739863065 0.980928951697069 2 36 "4.3" [] [[1.0, 0.0]] 2 true true false false -0.16666666666666666 0 2.44719771968788722276449401148 ["ModularForm/GL2/Q/holomorphic/36/3/d/c/19/2"] "2-6e2-4.3-c2-0-2" 0.9904185739863065 0.980928951697069 2 36 "4.3" [] [[1.0, 0.0]] 2 true true true true 0.0 0 3.42919429089720191179666185785 ["ModularForm/GL2/Q/holomorphic/36/3/d/b/19/1", "ModularForm/GL2/Q/holomorphic/36/3/d/b"] "2-6e2-4.3-c2-0-3" 0.9904185739863065 0.980928951697069 2 36 "4.3" [] [[1.0, 0.0]] 2 true true false false 0.16666666666666666 0 3.84845937326100545215129282723 ["ModularForm/GL2/Q/holomorphic/36/3/d/c/19/1"] "2-6e2-9.2-c2-0-0" 0.9904185739863065 0.980928951697069 2 36 "9.2" [] [[1.0, 0.0]] 2 true true false false -0.06543697156089089 0 2.50959671808882599528232416564 ["ModularForm/GL2/Q/holomorphic/36/3/g/a/29/2"] "2-6e2-9.2-c2-0-1" 0.9904185739863065 0.980928951697069 2 36 "9.2" [] [[1.0, 0.0]] 2 true true false false 0.12099252711644644 0 3.37995410589960562088165736252 ["ModularForm/GL2/Q/holomorphic/36/3/g/a/29/1"] "2-6e2-9.5-c2-0-0" 0.9904185739863065 0.980928951697069 2 36 "9.5" [] [[1.0, 0.0]] 2 true true false false -0.12099252711644644 0 2.07470438107883879783765653509 ["ModularForm/GL2/Q/holomorphic/36/3/g/a/5/1"] "2-6e2-9.5-c2-0-1" 0.9904185739863065 0.980928951697069 2 36 "9.5" [] [[1.0, 0.0]] 2 true true false false 0.06543697156089089 0 3.28606872193177282185519833530 ["ModularForm/GL2/Q/holomorphic/36/3/g/a/5/2"] "2-6e2-1.1-c3-0-0" 1.4574185260955974 2.1240687602066632 2 36 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.0 0 2.21712775913447454783926853802 ["ModularForm/GL2/Q/holomorphic/36/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/36/4/a/a"] "2-6e2-12.11-c3-0-0" 1.4574185260955974 2.1240687602066632 2 36 "12.11" [] [[1.5, 0.0]] 3 true true false false -0.23247398524751445 0 1.06914759206160677010192137974 ["ModularForm/GL2/Q/holomorphic/36/4/b/b/35/2"] "2-6e2-12.11-c3-0-1" 1.4574185260955974 2.1240687602066632 2 36 "12.11" [] [[1.5, 0.0]] 3 true true false false -0.3479566380076518 0 1.21845761788249504021230692632 ["ModularForm/GL2/Q/holomorphic/36/4/b/a/35/2"] "2-6e2-12.11-c3-0-2" 1.4574185260955974 2.1240687602066632 2 36 "12.11" [] [[1.5, 0.0]] 3 true true false false -0.07161273873718195 0 2.47557094283576162337809931225 ["ModularForm/GL2/Q/holomorphic/36/4/b/b/35/4"] "2-6e2-12.11-c3-0-3" 1.4574185260955974 2.1240687602066632 2 36 "12.11" [] [[1.5, 0.0]] 3 true true false false 0.23247398524751445 0 2.51366582009756001478481545201 ["ModularForm/GL2/Q/holomorphic/36/4/b/b/35/1"] "2-6e2-12.11-c3-0-4" 1.4574185260955974 2.1240687602066632 2 36 "12.11" [] [[1.5, 0.0]] 3 true true false false 0.07161273873718195 0 3.10471657988589878445929056958 ["ModularForm/GL2/Q/holomorphic/36/4/b/b/35/3"] "2-6e2-12.11-c3-0-5" 1.4574185260955974 2.1240687602066632 2 36 "12.11" [] [[1.5, 0.0]] 3 true true false false 0.3479566380076518 0 3.62431241708097699062982397093 ["ModularForm/GL2/Q/holomorphic/36/4/b/a/35/1"] "2-6e2-36.11-c3-0-0" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false 0.4426521545428372 0 0.34566135393146410948046424907 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/3"] "2-6e2-36.11-c3-0-1" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false -0.4608627773406481 0 0.64127928453103219404222147034 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/7"] "2-6e2-36.11-c3-0-10" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false 0.1097694040832802 0 2.76901326243391386402702945303 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/9"] "2-6e2-36.11-c3-0-11" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false 0.023355457657166127 0 3.04995245676196326254894860869 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/12"] "2-6e2-36.11-c3-0-12" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false 0.17403015367424893 0 3.49364060448882177283868948680 ["ModularForm/GL2/Q/holomorphic/36/4/h/a/11/4"] "2-6e2-36.11-c3-0-13" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false -0.49102994783270976 0 3.70462858021714604335693782027 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/1"] "2-6e2-36.11-c3-0-14" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false 0.29744683183793746 0 3.98669497740214899202011159451 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/8"] "2-6e2-36.11-c3-0-15" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false -0.468148595540429 0 4.47871680805328595193746332110 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/6"] "2-6e2-36.11-c3-0-2" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false -0.22197514266570587 0 0.846166483196409111940889925206 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/4"] "2-6e2-36.11-c3-0-3" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false 0.015471197759953563 0 1.55233714492890407054722128870 ["ModularForm/GL2/Q/holomorphic/36/4/h/a/11/1"] "2-6e2-36.11-c3-0-4" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false -0.2782463648073084 0 1.80820773089152819536429814431 ["ModularForm/GL2/Q/holomorphic/36/4/h/a/11/3"] "2-6e2-36.11-c3-0-5" 1.4574185260955974 2.1240687602066632 2 36 "36.11" [] [[1.5, 0.0]] 3 true true false false -0.09938695147840941 0 1.85488749605626474199913172940 ["ModularForm/GL2/Q/holomorphic/36/4/h/b/11/2"] 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["ModularForm/GL2/Q/holomorphic/36/33/g/a/5/16"] "2-6e2-9.5-c32-0-7" 15.281359838446798 233.51995851209477 2 36 "9.5" [] [[16.0, 0.0]] 32 true true false false -0.3512802451940013 0 0.30854161718493395842784468338 ["ModularForm/GL2/Q/holomorphic/36/33/g/a/5/27"] "2-6e2-9.5-c32-0-8" 15.281359838446798 233.51995851209477 2 36 "9.5" [] [[16.0, 0.0]] 32 true true false false 0.23114905667542324 0 0.31387156157601027288033601910 ["ModularForm/GL2/Q/holomorphic/36/33/g/a/5/15"] "2-6e2-9.5-c32-0-9" 15.281359838446798 233.51995851209477 2 36 "9.5" [] [[16.0, 0.0]] 32 true true false false -0.16311527568929865 0 0.35119505180124020858713624193 ["ModularForm/GL2/Q/holomorphic/36/33/g/a/5/31"] # 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).