Properties

Label 2-637-49.39-c1-0-21
Degree $2$
Conductor $637$
Sign $0.775 - 0.630i$
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  + (−0.157 − 0.0237i)2-s + (0.607 − 0.414i)3-s + (−1.88 − 0.582i)4-s + (0.234 + 3.13i)5-s + (−0.105 + 0.0508i)6-s + (1.73 − 1.99i)7-s + (0.570 + 0.274i)8-s + (−0.898 + 2.28i)9-s + (0.0373 − 0.498i)10-s + (−1.99 − 5.08i)11-s + (−1.38 + 0.428i)12-s + (0.623 − 0.781i)13-s + (−0.320 + 0.272i)14-s + (1.43 + 1.80i)15-s + (3.17 + 2.16i)16-s + (5.63 + 5.22i)17-s + ⋯
L(s)  = 1  + (−0.111 − 0.0167i)2-s + (0.350 − 0.239i)3-s + (−0.943 − 0.291i)4-s + (0.104 + 1.40i)5-s + (−0.0430 + 0.0207i)6-s + (0.656 − 0.754i)7-s + (0.201 + 0.0970i)8-s + (−0.299 + 0.763i)9-s + (0.0118 − 0.157i)10-s + (−0.601 − 1.53i)11-s + (−0.400 + 0.123i)12-s + (0.172 − 0.216i)13-s + (−0.0857 + 0.0729i)14-s + (0.371 + 0.466i)15-s + (0.794 + 0.541i)16-s + (1.36 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.775 - 0.630i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.775 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25640 + 0.446272i\)
\(L(\frac12)\) \(\approx\) \(1.25640 + 0.446272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.73 + 1.99i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (0.157 + 0.0237i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-0.607 + 0.414i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.234 - 3.13i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (1.99 + 5.08i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (-5.63 - 5.22i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-3.85 - 6.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.215 - 0.199i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (0.882 - 3.86i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.656 + 1.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.34 - 1.03i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (-1.95 - 0.941i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.0234 - 0.0113i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-11.2 - 1.69i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-7.87 - 2.42i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.551 - 7.35i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (7.39 - 2.28i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-3.53 + 6.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.33 + 5.84i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.46 + 0.371i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (6.32 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.52 - 6.92i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.86 + 4.74i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 9.35T + 97T^{2} \)
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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

−10.42932414309717562443529393258, −10.28151463135510476105035522260, −8.725153986196501492871763939519, −7.914966148168593100592529622385, −7.57660179609810026859216134996, −5.93056302985621104408710047975, −5.44946069396338293626875316193, −3.81258467023143849884748169522, −3.10771423361935471604761512280, −1.38110791026349001046141028178, 0.879782606072371666083745707827, 2.64544825485880789588894043861, 4.12442681989263783016010290899, 5.06605162212787214002955774264, 5.34996354053730214963230245785, 7.28204839619476160031213476523, 8.079343685377998117055987445620, 9.043981811901285760554382901993, 9.278042117923106382655363781609, 10.00589132789158104268078782671

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