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Properties

Label 2-637-91.5-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.0201 - 0.999i$
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  + (−0.531 − 1.98i)2-s + (1.57 − 0.909i)3-s + (−1.91 + 1.10i)4-s + (−2.75 + 0.737i)5-s + (−2.64 − 2.64i)6-s + (0.311 + 0.311i)8-s + (0.155 − 0.269i)9-s + (2.92 + 5.06i)10-s + (0.165 − 0.616i)11-s + (−2.01 + 3.48i)12-s + (−3.40 − 1.19i)13-s + (−3.66 + 3.66i)15-s + (−1.76 + 3.05i)16-s + (−2.16 − 3.74i)17-s + (−0.616 − 0.165i)18-s + (−4.64 + 1.24i)19-s + ⋯
L(s)  = 1  + (−0.375 − 1.40i)2-s + (0.909 − 0.525i)3-s + (−0.958 + 0.553i)4-s + (−1.23 + 0.329i)5-s + (−1.07 − 1.07i)6-s + (0.109 + 0.109i)8-s + (0.0518 − 0.0898i)9-s + (0.924 + 1.60i)10-s + (0.0498 − 0.186i)11-s + (−0.581 + 1.00i)12-s + (−0.943 − 0.330i)13-s + (−0.946 + 0.946i)15-s + (−0.440 + 0.763i)16-s + (−0.524 − 0.908i)17-s + (−0.145 − 0.0389i)18-s + (−1.06 + 0.285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0201 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0201 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0201 - 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0201 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215958 + 0.220360i\)
\(L(\frac12)\) \(\approx\) \(0.215958 + 0.220360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.40 + 1.19i)T \)
good2 \( 1 + (0.531 + 1.98i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-1.57 + 0.909i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.75 - 0.737i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.165 + 0.616i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.64 - 1.24i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.808 + 0.466i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 + (2.00 - 7.48i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.92 - 0.783i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.81 + 1.81i)T + 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (2.16 + 8.07i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.68 + 2.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.349 + 0.0935i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.74 - 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.94 + 2.66i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.56 + 5.56i)T - 71iT^{2} \)
73 \( 1 + (12.1 + 3.24i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.89 + 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.30 + 4.30i)T + 83iT^{2} \)
89 \( 1 + (2.05 + 7.66i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.236 - 0.236i)T + 97iT^{2} \)
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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

−10.22438403123804372279635218443, −8.902508528012831086432518319539, −8.509880814981958253874505839434, −7.50529752138546315772130763193, −6.76154817200000622208003666214, −4.85212258112814083073186132072, −3.66322795283805699903109380092, −2.90620802106121734064118865834, −1.99190212843982936916214298552, −0.15883350637808892436703008504, 2.64382322750597715015180970497, 4.10081710542136457096539276549, 4.63415196332307998138024970827, 6.11640801110986567508979415468, 6.99111757296989883201176360141, 8.025622564684149962599126306038, 8.291459273817769813271360602193, 9.150530101224128426574570221417, 9.877116999777325497519858429980, 11.18305575586248823559457313888

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