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Properties

Label 2-637-91.73-c1-0-1
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: $\chi_{637} (619, \cdot )$
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

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

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