Properties

Label 2-637-91.73-c1-0-9
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

\(L(1)\) \(\approx\) \(0.797824 + 0.781885i\)
\(L(\frac12)\) \(\approx\) \(0.797824 + 0.781885i\)
\(L(\frac{3}{2})\) 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

−10.62539258336121809503518983076, −9.738003551975661129401507602753, −8.968737830720353856232806846058, −7.919022487436442870330404127033, −6.94803821856912180813218039833, −6.39092616150274544248204688174, −5.67915661569107792046591069860, −5.10030216259398882181777524210, −2.99487735172443811442471023095, −1.14395953449459773342742429402, 1.01905074162542425071061284722, 2.17867533620045824118885712359, 3.53024693296831700462352935711, 4.70419740169491450233636249421, 5.80021082034667034706351028394, 6.33064693511059764452510429238, 8.142861238385380967228006327056, 9.157265021000377870835290916076, 9.774799175717311033336006014693, 10.48780281824213524967221471186

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