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Properties

Label 2-637-13.9-c1-0-25
Degree $2$
Conductor $637$
Sign $0.997 - 0.0638i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.929 + 1.60i)2-s + (−1.14 + 1.98i)3-s + (−0.726 − 1.25i)4-s − 0.197·5-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 + 3.33·12-s + (−2.72 − 2.36i)13-s + (0.226 − 0.392i)15-s + (2.39 − 4.15i)16-s + (−0.420 − 0.728i)17-s + 4.20·18-s + (−0.675 − 1.17i)19-s + ⋯
L(s)  = 1  + (−0.656 + 1.13i)2-s + (−0.662 + 1.14i)3-s + (−0.363 − 0.629i)4-s − 0.0882·5-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 + 0.962·12-s + (−0.755 − 0.655i)13-s + (0.0584 − 0.101i)15-s + (0.599 − 1.03i)16-s + (−0.102 − 0.176i)17-s + 0.991·18-s + (−0.155 − 0.268i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.997 - 0.0638i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.997 - 0.0638i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337142 + 0.0107672i\)
\(L(\frac12)\) \(\approx\) \(0.337142 + 0.0107672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (2.72 + 2.36i)T \)
good2 \( 1 + (0.929 - 1.60i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.14 - 1.98i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.197T + 5T^{2} \)
11 \( 1 + (-2.09 + 3.62i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.420 + 0.728i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.675 + 1.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.05 + 3.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.11 + 7.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.28T + 31T^{2} \)
37 \( 1 + (1.52 - 2.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.69 - 4.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.66 + 4.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 4.64T + 53T^{2} \)
59 \( 1 + (3.02 + 5.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.68 - 9.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.69 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.98 - 5.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.88T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 3.07T + 83T^{2} \)
89 \( 1 + (-5.99 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.73 + 16.8i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20830046578105199600210825037, −9.764534237548410500658802883807, −8.736699752844425021778693722228, −8.098323127218814549729461694287, −6.97095529371462665487716466358, −6.07625166947080600980723001809, −5.35571779068266942727777109476, −4.32324939539361719622541471875, −3.04892806531106091612971625469, −0.25451494206916139887286050188, 1.42096509711317658192033369289, 2.09719734668459183692693502641, 3.65184674771719494053593861782, 5.06417880846128309809437828047, 6.36374730901432109780482957369, 7.00258435204505576443813500166, 7.943170121072919682389682103829, 9.168542659255270562366005310476, 9.738516749194709021243573693572, 10.71066061276649844186598359675

Graph of the $Z$-function along the critical line