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Properties

Label 2-637-7.4-c1-0-20
Degree $2$
Conductor $637$
Sign $0.947 - 0.318i$
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  + (0.328 + 0.568i)2-s + (0.102 − 0.177i)3-s + (0.784 − 1.35i)4-s + (0.679 + 1.17i)5-s + 0.134·6-s + 2.34·8-s + (1.47 + 2.56i)9-s + (−0.446 + 0.772i)10-s + (0.952 − 1.65i)11-s + (−0.160 − 0.278i)12-s + 13-s + 0.278·15-s + (−0.800 − 1.38i)16-s + (−1.78 + 3.08i)17-s + (−0.970 + 1.68i)18-s + (−0.492 − 0.853i)19-s + ⋯
L(s)  = 1  + (0.231 + 0.401i)2-s + (0.0591 − 0.102i)3-s + (0.392 − 0.679i)4-s + (0.304 + 0.526i)5-s + 0.0548·6-s + 0.828·8-s + (0.493 + 0.853i)9-s + (−0.141 + 0.244i)10-s + (0.287 − 0.497i)11-s + (−0.0463 − 0.0803i)12-s + 0.277·13-s + 0.0718·15-s + (−0.200 − 0.346i)16-s + (−0.432 + 0.749i)17-s + (−0.228 + 0.396i)18-s + (−0.113 − 0.195i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12497 + 0.347530i\)
\(L(\frac12)\) \(\approx\) \(2.12497 + 0.347530i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (-0.328 - 0.568i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.102 + 0.177i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.679 - 1.17i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.952 + 1.65i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.78 - 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.492 + 0.853i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.848 + 1.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.54T + 29T^{2} \)
31 \( 1 + (3.84 - 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.01 - 1.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.88T + 41T^{2} \)
43 \( 1 + 3.16T + 43T^{2} \)
47 \( 1 + (3.88 + 6.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.177 - 0.306i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.08 - 1.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.10 + 10.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.65 - 9.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.05T + 71T^{2} \)
73 \( 1 + (-3.56 + 6.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.69 - 4.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 + (-3.44 - 5.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−10.62561640143010208439573870431, −10.04655207863986345936528155736, −8.790165690490013739313562936043, −7.86190457149334874986839373063, −6.77904200944070560392274593739, −6.32485802986391215105583671072, −5.25026103693776246341549554763, −4.26942098262410053137293070000, −2.68141291035781260462684965015, −1.50231218827177621438038059862, 1.41226558446146016876854709612, 2.76326822632018647138816443641, 3.94053028275811712425050080207, 4.67954413496777973260434484352, 6.08726997742657281769606206600, 7.02907148231159407476500726971, 7.84203088891088240467489245030, 9.015645321429733422460718217558, 9.558833375308876548933023740099, 10.64669519257224733813597042567

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