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Properties

Label 2-637-91.81-c1-0-1
Degree $2$
Conductor $637$
Sign $-0.828 - 0.559i$
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  + (−1.11 + 1.92i)2-s − 0.549·3-s + (−1.46 − 2.53i)4-s + (−2.11 − 3.65i)5-s + (0.610 − 1.05i)6-s + 2.06·8-s − 2.69·9-s + 9.36·10-s − 0.549·11-s + (0.804 + 1.39i)12-s + (2.95 + 2.06i)13-s + (1.15 + 2.00i)15-s + (0.640 − 1.10i)16-s + (−1.18 − 2.06i)17-s + (2.99 − 5.18i)18-s + 3.61·19-s + ⋯
L(s)  = 1  + (−0.784 + 1.35i)2-s − 0.317·3-s + (−0.732 − 1.26i)4-s + (−0.943 − 1.63i)5-s + (0.249 − 0.431i)6-s + 0.728·8-s − 0.899·9-s + 2.96·10-s − 0.165·11-s + (0.232 + 0.402i)12-s + (0.820 + 0.571i)13-s + (0.299 + 0.518i)15-s + (0.160 − 0.277i)16-s + (−0.288 − 0.499i)17-s + (0.705 − 1.22i)18-s + 0.828·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.828 - 0.559i$
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: \((2,\ 637,\ (\ :1/2),\ -0.828 - 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100510 + 0.328711i\)
\(L(\frac12)\) \(\approx\) \(0.100510 + 0.328711i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.95 - 2.06i)T \)
good2 \( 1 + (1.11 - 1.92i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 0.549T + 3T^{2} \)
5 \( 1 + (2.11 + 3.65i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.549T + 11T^{2} \)
17 \( 1 + (1.18 + 2.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.61T + 19T^{2} \)
23 \( 1 + (2.90 - 5.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.79 - 3.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.57 - 4.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.164 + 0.285i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.14 + 5.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.61 - 2.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.10 - 7.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.32 - 2.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.903 + 1.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 0.609T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.45 + 4.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.00 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + (3.73 - 6.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.42 - 5.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.95667408223139549159178841800, −9.385367756587852825629836451748, −9.037904319696806645610399061512, −8.224902746888293577307586653763, −7.67950143319901477219792622066, −6.58801981772831476824636644206, −5.47361004929429703732044759832, −5.01059389685384666989733510232, −3.62642988455527523099939148574, −1.05081821554537837210656192283, 0.31400347664675208899404285485, 2.38169017320487844417861514799, 3.20072158448127151125207929001, 3.97543281776679977272822138574, 5.84142570818489195999888120862, 6.73345341220008285906169184281, 8.023675610206806497534341635965, 8.407504996725284637616001061655, 9.758585221491163607806057455213, 10.56853179221189628524185386754

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