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Properties

Label 2-637-13.4-c1-0-24
Degree $2$
Conductor $637$
Sign $0.711 - 0.702i$
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  + (1.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + 1.73i·5-s + (1.5 − 0.866i)6-s − 1.73i·8-s + (1 + 1.73i)9-s + (−1.49 + 2.59i)10-s + (4.5 + 2.59i)11-s + 12-s + (−1 + 3.46i)13-s + (1.49 + 0.866i)15-s + (2.49 − 4.33i)16-s + (−3 − 5.19i)17-s + 3.46i·18-s + (1.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s + 0.774i·5-s + (0.612 − 0.353i)6-s − 0.612i·8-s + (0.333 + 0.577i)9-s + (−0.474 + 0.821i)10-s + (1.35 + 0.783i)11-s + 0.288·12-s + (−0.277 + 0.960i)13-s + (0.387 + 0.223i)15-s + (0.624 − 1.08i)16-s + (−0.727 − 1.26i)17-s + 0.816i·18-s + (0.344 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69385 + 1.10572i\)
\(L(\frac12)\) \(\approx\) \(2.69385 + 1.10572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.66iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3 + 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.81465526077469446187485662760, −9.693273809155868501962119628376, −9.018532995973055306697499621477, −7.46852369968244475550418428448, −6.88284436342934848642114399837, −6.55734967284248223953985967840, −5.02394722973682661881662539021, −4.37979521354408581843519055815, −3.15308784196051491565154291431, −1.78462523077700191972086828515, 1.39218443189360347516147319258, 3.09067155721651091731584967940, 3.88683415262428967646661788503, 4.57021706512070215732170847634, 5.65996802633039328994567113757, 6.55022966253346916898598805457, 8.193318940468638333753638078889, 8.738198424382278938789984073035, 9.658634634861334117246218853835, 10.63466422616668553527388451937

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