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Properties

Label 2-637-91.74-c1-0-31
Degree $2$
Conductor $637$
Sign $0.788 - 0.615i$
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  + 2.61·2-s + (1.30 + 2.26i)3-s + 4.85·4-s + (−1.30 − 2.26i)5-s + (3.42 + 5.93i)6-s + 7.47·8-s + (−1.92 + 3.33i)9-s + (−3.42 − 5.93i)10-s + (−0.927 − 1.60i)11-s + (6.35 + 11.0i)12-s + (−2.5 + 2.59i)13-s + (3.42 − 5.93i)15-s + 9.85·16-s − 1.47·17-s + (−5.04 + 8.73i)18-s + (0.927 − 1.60i)19-s + ⋯
L(s)  = 1  + 1.85·2-s + (0.755 + 1.30i)3-s + 2.42·4-s + (−0.585 − 1.01i)5-s + (1.39 + 2.42i)6-s + 2.64·8-s + (−0.642 + 1.11i)9-s + (−1.08 − 1.87i)10-s + (−0.279 − 0.484i)11-s + (1.83 + 3.17i)12-s + (−0.693 + 0.720i)13-s + (0.884 − 1.53i)15-s + 2.46·16-s − 0.357·17-s + (−1.18 + 2.05i)18-s + (0.212 − 0.368i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.55554 + 1.56846i\)
\(L(\frac12)\) \(\approx\) \(4.55554 + 1.56846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 + (-1.30 - 2.26i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.30 + 2.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.927 + 1.60i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 + (-0.927 + 1.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + (3.54 - 6.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.35 + 4.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (0.381 - 0.661i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.28 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.11 - 1.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.88 - 3.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.23T + 59T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.35 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 - 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 - 4.90T + 89T^{2} \)
97 \( 1 + (9.42 + 16.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.94433054022492507375361283342, −9.911369940519728172514554061191, −8.940124340853274303994246243647, −8.053790102898292210512235183565, −6.91256867215572089599375996086, −5.57029944182178105596647264099, −4.79403713733497976918580727950, −4.21395319475268831688290622089, −3.47515071161759657347342510798, −2.32351608880879227192380520532, 2.03914034028436336896142654439, 2.81402447053182029100347526799, 3.62992044980620075378954273268, 4.86545141379106381188143809306, 6.13898392007460558667063029527, 6.79252910823150672822878048793, 7.62349574484209797396147154989, 7.983638427302731415471483714642, 9.895930084054878414588431388709, 10.94414540142417443506037448172

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