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Properties

Label 2-637-91.23-c1-0-36
Degree 22
Conductor 637637
Sign −0.372+0.927i-0.372 + 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+0.927i-0.372 + 0.927i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(569,⋅)\chi_{637} (569, \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 1.28034−1.89444i1.28034 - 1.89444i
L(12)L(\frac12) ≈\approx 1.28034−1.89444i1.28034 - 1.89444i
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

−10.25456436509819797530968980076, −9.438330609683080154154176879487, −8.296139100345401163773963796824, −7.65603876875911550180234183376, −6.73026388398757019681449655512, −6.39050970461606434915508643514, −4.56559258033460906841430322472, −3.10690378896087616522524560119, −2.26865212919591692733733551697, −1.24420679570262469178396602579, 2.20527056451832008666077908539, 3.29720734956952129651356291656, 4.18422274463419537881382882272, 5.28170885232712290420499610630, 6.42883799761839721886692724428, 7.37334985932552469135494613465, 8.394337084916896903891577095202, 9.088761147536696557379497046712, 9.995010186324360782261770556849, 10.65113141764913839070341471393

Graph of the ZZ-function along the critical line