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Properties

Label 2-637-91.4-c1-0-23
Degree 22
Conductor 637637
Sign 0.372+0.927i0.372 + 0.927i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.456i·2-s + (−1.39 − 2.41i)3-s + 1.79·4-s + (0.395 − 0.228i)5-s + (1.10 − 0.637i)6-s + 1.73i·8-s + (−2.39 + 4.14i)9-s + (0.104 + 0.180i)10-s + (3.39 − 1.96i)11-s + (−2.5 − 4.33i)12-s + (3.5 − 0.866i)13-s + (−1.10 − 0.637i)15-s + 2.79·16-s + 3·17-s + (−1.89 − 1.09i)18-s + (−1.18 − 0.685i)19-s + ⋯
L(s)  = 1  + 0.323i·2-s + (−0.805 − 1.39i)3-s + 0.895·4-s + (0.176 − 0.102i)5-s + (0.450 − 0.260i)6-s + 0.612i·8-s + (−0.798 + 1.38i)9-s + (0.0330 + 0.0571i)10-s + (1.02 − 0.591i)11-s + (−0.721 − 1.25i)12-s + (0.970 − 0.240i)13-s + (−0.285 − 0.164i)15-s + 0.697·16-s + 0.727·17-s + (−0.446 − 0.257i)18-s + (−0.272 − 0.157i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.372+0.927i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.372+0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.372+0.927i0.372 + 0.927i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(459,⋅)\chi_{637} (459, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.372+0.927i)(2,\ 637,\ (\ :1/2),\ 0.372 + 0.927i)

Particular Values

L(1)L(1) ≈\approx 1.29389−0.874471i1.29389 - 0.874471i
L(12)L(\frac12) ≈\approx 1.29389−0.874471i1.29389 - 0.874471i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
13 1+(−3.5+0.866i)T 1 + (-3.5 + 0.866i)T
good2 1−0.456iT−2T2 1 - 0.456iT - 2T^{2}
3 1+(1.39+2.41i)T+(−1.5+2.59i)T2 1 + (1.39 + 2.41i)T + (-1.5 + 2.59i)T^{2}
5 1+(−0.395+0.228i)T+(2.5−4.33i)T2 1 + (-0.395 + 0.228i)T + (2.5 - 4.33i)T^{2}
11 1+(−3.39+1.96i)T+(5.5−9.52i)T2 1 + (-3.39 + 1.96i)T + (5.5 - 9.52i)T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+(1.18+0.685i)T+(9.5+16.4i)T2 1 + (1.18 + 0.685i)T + (9.5 + 16.4i)T^{2}
23 1+1.58T+23T2 1 + 1.58T + 23T^{2}
29 1+(−3.39+5.88i)T+(−14.5−25.1i)T2 1 + (-3.39 + 5.88i)T + (-14.5 - 25.1i)T^{2}
31 1+(7.5+4.33i)T+(15.5+26.8i)T2 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2}
37 1+6.92iT−37T2 1 + 6.92iT - 37T^{2}
41 1+(−6.79−3.92i)T+(20.5+35.5i)T2 1 + (-6.79 - 3.92i)T + (20.5 + 35.5i)T^{2}
43 1+(4.68+8.11i)T+(−21.5+37.2i)T2 1 + (4.68 + 8.11i)T + (-21.5 + 37.2i)T^{2}
47 1+(8.29−4.78i)T+(23.5−40.7i)T2 1 + (8.29 - 4.78i)T + (23.5 - 40.7i)T^{2}
53 1+(3.08−5.33i)T+(−26.5−45.8i)T2 1 + (3.08 - 5.33i)T + (-26.5 - 45.8i)T^{2}
59 1−12.3iT−59T2 1 - 12.3iT - 59T^{2}
61 1+(−7.37+12.7i)T+(−30.5−52.8i)T2 1 + (-7.37 + 12.7i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.87+2.23i)T+(33.5−58.0i)T2 1 + (-3.87 + 2.23i)T + (33.5 - 58.0i)T^{2}
71 1+(−3.79+2.18i)T+(35.5−61.4i)T2 1 + (-3.79 + 2.18i)T + (35.5 - 61.4i)T^{2}
73 1+(−3−1.73i)T+(36.5+63.2i)T2 1 + (-3 - 1.73i)T + (36.5 + 63.2i)T^{2}
79 1+(−3−5.19i)T+(−39.5+68.4i)T2 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2}
83 1+7.02iT−83T2 1 + 7.02iT - 83T^{2}
89 1−16.1iT−89T2 1 - 16.1iT - 89T^{2}
97 1+(6.31−3.64i)T+(48.5−84.0i)T2 1 + (6.31 - 3.64i)T + (48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.93255889481806058243812868331, −9.501379518986480841069363609490, −8.258185678958339729471424206629, −7.60432479327146753986803404896, −6.70849290467544021938648239693, −6.02754290170039811582021070218, −5.58091600014260064667609010277, −3.65128888613961709687098563126, −2.08721821007536886643836102810, −1.06001833414365627601565033938, 1.56922570399140501144258352674, 3.33473671278930713498370274384, 4.06722926122427706905716311439, 5.23009486497388365542472260837, 6.23294361841214697552853423746, 6.82152337185676039283662164618, 8.299408809323139093153817896978, 9.452764038797859625091863957229, 10.06306393533723756994617105063, 10.72434972235414617573360021568

Graph of the ZZ-function along the critical line