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Properties

Label 2-637-91.16-c1-0-38
Degree 22
Conductor 637637
Sign 0.326+0.945i0.326 + 0.945i
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  + 1.85·2-s + (1.14 − 1.98i)3-s + 1.45·4-s + (−0.0986 + 0.170i)5-s + (2.13 − 3.69i)6-s − 1.01·8-s + (−1.13 − 1.95i)9-s + (−0.183 + 0.317i)10-s + (2.09 − 3.62i)11-s + (1.66 − 2.88i)12-s + (2.72 − 2.36i)13-s + (0.226 + 0.392i)15-s − 4.79·16-s − 0.841·17-s + (−2.10 − 3.64i)18-s + (0.675 + 1.17i)19-s + ⋯
L(s)  = 1  + 1.31·2-s + (0.662 − 1.14i)3-s + 0.726·4-s + (−0.0441 + 0.0764i)5-s + (0.870 − 1.50i)6-s − 0.359·8-s + (−0.377 − 0.653i)9-s + (−0.0579 + 0.100i)10-s + (0.630 − 1.09i)11-s + (0.481 − 0.833i)12-s + (0.755 − 0.655i)13-s + (0.0584 + 0.101i)15-s − 1.19·16-s − 0.204·17-s + (−0.495 − 0.858i)18-s + (0.155 + 0.268i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.326+0.945i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.326+0.945i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.326+0.945i0.326 + 0.945i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(471,⋅)\chi_{637} (471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.326+0.945i)(2,\ 637,\ (\ :1/2),\ 0.326 + 0.945i)

Particular Values

L(1)L(1) ≈\approx 2.76310−1.96802i2.76310 - 1.96802i
L(12)L(\frac12) ≈\approx 2.76310−1.96802i2.76310 - 1.96802i
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+(−2.72+2.36i)T 1 + (-2.72 + 2.36i)T
good2 1−1.85T+2T2 1 - 1.85T + 2T^{2}
3 1+(−1.14+1.98i)T+(−1.5−2.59i)T2 1 + (-1.14 + 1.98i)T + (-1.5 - 2.59i)T^{2}
5 1+(0.0986−0.170i)T+(−2.5−4.33i)T2 1 + (0.0986 - 0.170i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.09+3.62i)T+(−5.5−9.52i)T2 1 + (-2.09 + 3.62i)T + (-5.5 - 9.52i)T^{2}
17 1+0.841T+17T2 1 + 0.841T + 17T^{2}
19 1+(−0.675−1.17i)T+(−9.5+16.4i)T2 1 + (-0.675 - 1.17i)T + (-9.5 + 16.4i)T^{2}
23 1+4.11T+23T2 1 + 4.11T + 23T^{2}
29 1+(−4.11−7.13i)T+(−14.5+25.1i)T2 1 + (-4.11 - 7.13i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.640+1.10i)T+(−15.5+26.8i)T2 1 + (0.640 + 1.10i)T + (-15.5 + 26.8i)T^{2}
37 1−3.04T+37T2 1 - 3.04T + 37T^{2}
41 1+(−2.69−4.67i)T+(−20.5+35.5i)T2 1 + (-2.69 - 4.67i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.66−4.61i)T+(−21.5−37.2i)T2 1 + (2.66 - 4.61i)T + (-21.5 - 37.2i)T^{2}
47 1+(5.83−10.1i)T+(−23.5−40.7i)T2 1 + (5.83 - 10.1i)T + (-23.5 - 40.7i)T^{2}
53 1+(2.32+4.02i)T+(−26.5+45.8i)T2 1 + (2.32 + 4.02i)T + (-26.5 + 45.8i)T^{2}
59 1+6.05T+59T2 1 + 6.05T + 59T^{2}
61 1+(5.68+9.84i)T+(−30.5+52.8i)T2 1 + (5.68 + 9.84i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.69−11.6i)T+(−33.5−58.0i)T2 1 + (6.69 - 11.6i)T + (-33.5 - 58.0i)T^{2}
71 1+(−2.98+5.17i)T+(−35.5−61.4i)T2 1 + (-2.98 + 5.17i)T + (-35.5 - 61.4i)T^{2}
73 1+(−1.94−3.36i)T+(−36.5+63.2i)T2 1 + (-1.94 - 3.36i)T + (-36.5 + 63.2i)T^{2}
79 1+(−5.36+9.29i)T+(−39.5−68.4i)T2 1 + (-5.36 + 9.29i)T + (-39.5 - 68.4i)T^{2}
83 1+3.07T+83T2 1 + 3.07T + 83T^{2}
89 1−11.9T+89T2 1 - 11.9T + 89T^{2}
97 1+(−9.73+16.8i)T+(−48.5−84.0i)T2 1 + (-9.73 + 16.8i)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

−10.80299919888429879540466033297, −9.285364602641729560173425186695, −8.473347051660821810571517236639, −7.70398477176143873328067776235, −6.48482440125680120288174421588, −6.07070325171585040219128455524, −4.83566968034486254794606180293, −3.50589051999421539977626693469, −2.92934278500421617752703630122, −1.35379884565889016145538646438, 2.28768504223823744189439885375, 3.52683143997779211880704356583, 4.28984917111468093101783385148, 4.71266604731963780440994132550, 6.05258996623747266684443541089, 6.87880661546915129576860941141, 8.347550413156564810009952289595, 9.157619770806648842696099015142, 9.806744252244703481406170170786, 10.76589926642652659522782672458

Graph of the ZZ-function along the critical line