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Properties

Label 2-637-13.10-c1-0-42
Degree $2$
Conductor $637$
Sign $-0.980 - 0.195i$
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  + (2.30 − 1.33i)2-s + (−1.59 − 2.76i)3-s + (2.54 − 4.40i)4-s − 0.329i·5-s + (−7.35 − 4.24i)6-s − 8.21i·8-s + (−3.58 + 6.21i)9-s + (−0.438 − 0.759i)10-s + (0.411 − 0.237i)11-s − 16.2·12-s + (2.74 + 2.34i)13-s + (−0.909 + 0.525i)15-s + (−5.84 − 10.1i)16-s + (0.626 − 1.08i)17-s + 19.0i·18-s + (3.06 + 1.76i)19-s + ⋯
L(s)  = 1  + (1.62 − 0.941i)2-s + (−0.920 − 1.59i)3-s + (1.27 − 2.20i)4-s − 0.147i·5-s + (−3.00 − 1.73i)6-s − 2.90i·8-s + (−1.19 + 2.07i)9-s + (−0.138 − 0.240i)10-s + (0.124 − 0.0716i)11-s − 4.68·12-s + (0.760 + 0.649i)13-s + (−0.234 + 0.135i)15-s + (−1.46 − 2.52i)16-s + (0.151 − 0.263i)17-s + 4.50i·18-s + (0.702 + 0.405i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270514 + 2.74194i\)
\(L(\frac12)\) \(\approx\) \(0.270514 + 2.74194i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.74 - 2.34i)T \)
good2 \( 1 + (-2.30 + 1.33i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.59 + 2.76i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.329iT - 5T^{2} \)
11 \( 1 + (-0.411 + 0.237i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.626 + 1.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.06 - 1.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.661 - 1.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.96 + 5.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.85iT - 31T^{2} \)
37 \( 1 + (4.04 - 2.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.89 - 2.82i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.42 - 4.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.30iT - 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + (1.48 + 0.857i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.91 - 5.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.47 + 4.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.6 - 6.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.20iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 8.18iT - 83T^{2} \)
89 \( 1 + (-9.43 + 5.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.0 + 7.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.82836154056581860222800109675, −9.615761618152483222562264437990, −8.035408075308064256016553201767, −6.90692751122821978419312949521, −6.26603021783569262350759671957, −5.54038360972850209377057271404, −4.65796149928034984116303517723, −3.27610566838716632833617747887, −1.98858521929131227280654393425, −1.09114977163995223856841142973, 3.26839497744223925632621806065, 3.69004222577561387500622983510, 5.01181774374044182863539754474, 5.18578588111499647875341240459, 6.23929470400917323565704565974, 6.94718617137509830942598057254, 8.312387132100969139910867967846, 9.293246224671313260691276485669, 10.66787229880689186374463181508, 10.98407960380905660924222238730

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