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Properties

Label 2-637-91.81-c1-0-6
Degree $2$
Conductor $637$
Sign $-0.962 + 0.270i$
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  + (−1.15 + 1.99i)2-s − 2.16·3-s + (−1.65 − 2.86i)4-s + (1.08 + 1.87i)5-s + (2.49 − 4.32i)6-s + 2.99·8-s + 1.69·9-s − 4.99·10-s + 4.90·11-s + (3.57 + 6.19i)12-s + (1.41 + 3.31i)13-s + (−2.34 − 4.06i)15-s + (−0.151 + 0.262i)16-s + (3.57 + 6.19i)17-s + (−1.95 + 3.38i)18-s − 2.16·19-s + ⋯
L(s)  = 1  + (−0.814 + 1.41i)2-s − 1.25·3-s + (−0.825 − 1.43i)4-s + (0.484 + 0.839i)5-s + (1.01 − 1.76i)6-s + 1.06·8-s + 0.565·9-s − 1.57·10-s + 1.47·11-s + (1.03 + 1.78i)12-s + (0.391 + 0.920i)13-s + (−0.606 − 1.05i)15-s + (−0.0378 + 0.0655i)16-s + (0.868 + 1.50i)17-s + (−0.460 + 0.797i)18-s − 0.497·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.962 + 0.270i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0817953 - 0.594009i\)
\(L(\frac12)\) \(\approx\) \(0.0817953 - 0.594009i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 3.31i)T \)
good2 \( 1 + (1.15 - 1.99i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 2.16T + 3T^{2} \)
5 \( 1 + (-1.08 - 1.87i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 4.90T + 11T^{2} \)
17 \( 1 + (-3.57 - 6.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.16T + 19T^{2} \)
23 \( 1 + (-0.302 + 0.524i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.15 - 1.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.57 - 6.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.30 - 7.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.99 + 8.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.25 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.755 + 1.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.19 - 2.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.41 + 2.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 4.33T + 61T^{2} \)
67 \( 1 - T + 67T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.16 - 3.75i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.30 - 5.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + (-3.25 + 5.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.83 - 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.62038186654454957949196322665, −10.30890352033288525190082313754, −9.073285094618360481153253458874, −8.506471401017306110656349133921, −7.06767004553190363969496431811, −6.55923408345641586377799596268, −6.12028226255572705860102289384, −5.21617345683388593996720415946, −3.77814706079153294699345162486, −1.43321711011207616419629326184, 0.59491316310844949994328576582, 1.45489786088699784137537782638, 3.09727931737208258718285322023, 4.39952828842504443741813713425, 5.49249755285042169426601668523, 6.30057653774844438830171170188, 7.71423354379657254041901644199, 8.880872867274206210603272130993, 9.443137761428709148920442236705, 10.15157169401301630181984935412

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