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Properties

Label 2-637-91.83-c1-0-12
Degree $2$
Conductor $637$
Sign $0.895 - 0.445i$
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  + (1.15 − 1.15i)2-s + 1.98i·3-s − 0.678i·4-s + (−2.02 − 2.02i)5-s + (2.29 + 2.29i)6-s + (1.52 + 1.52i)8-s − 0.925·9-s − 4.69·10-s + (2.44 + 2.44i)11-s + 1.34·12-s + (−2.29 + 2.78i)13-s + (4.02 − 4.02i)15-s + 4.89·16-s + 6.44·17-s + (−1.07 + 1.07i)18-s + (2.14 + 2.14i)19-s + ⋯
L(s)  = 1  + (0.818 − 0.818i)2-s + 1.14i·3-s − 0.339i·4-s + (−0.907 − 0.907i)5-s + (0.936 + 0.936i)6-s + (0.540 + 0.540i)8-s − 0.308·9-s − 1.48·10-s + (0.737 + 0.737i)11-s + 0.387·12-s + (−0.635 + 0.771i)13-s + (1.03 − 1.03i)15-s + 1.22·16-s + 1.56·17-s + (−0.252 + 0.252i)18-s + (0.493 + 0.493i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06053 + 0.484539i\)
\(L(\frac12)\) \(\approx\) \(2.06053 + 0.484539i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 - 2.78i)T \)
good2 \( 1 + (-1.15 + 1.15i)T - 2iT^{2} \)
3 \( 1 - 1.98iT - 3T^{2} \)
5 \( 1 + (2.02 + 2.02i)T + 5iT^{2} \)
11 \( 1 + (-2.44 - 2.44i)T + 11iT^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + (-2.14 - 2.14i)T + 19iT^{2} \)
23 \( 1 - 3.43iT - 23T^{2} \)
29 \( 1 + 4.40T + 29T^{2} \)
31 \( 1 + (2.37 + 2.37i)T + 31iT^{2} \)
37 \( 1 + (2.75 + 2.75i)T + 37iT^{2} \)
41 \( 1 + (-3.03 - 3.03i)T + 41iT^{2} \)
43 \( 1 + 4.48iT - 43T^{2} \)
47 \( 1 + (-4.03 + 4.03i)T - 47iT^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (-2.09 + 2.09i)T - 59iT^{2} \)
61 \( 1 + 3.50iT - 61T^{2} \)
67 \( 1 + (-5.15 + 5.15i)T - 67iT^{2} \)
71 \( 1 + (-8.31 + 8.31i)T - 71iT^{2} \)
73 \( 1 + (-3.30 + 3.30i)T - 73iT^{2} \)
79 \( 1 - 1.08T + 79T^{2} \)
83 \( 1 + (2.01 + 2.01i)T + 83iT^{2} \)
89 \( 1 + (3.50 - 3.50i)T - 89iT^{2} \)
97 \( 1 + (-3.20 - 3.20i)T + 97iT^{2} \)
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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

−10.84605857778920313768046428265, −9.727886078253635875315341778352, −9.345988583538703813683422695052, −8.031783884082416684442164961370, −7.30721023729764147330650741430, −5.43507297961771934891675962349, −4.77206868931942391777627674341, −3.95271087422336672806058021564, −3.51046805579788860610668729597, −1.67418504344615553951321196504, 1.03084242150654640847134035462, 2.98748822069190093587460625463, 3.90553072111930931529266326805, 5.28272187728872954813341611251, 6.17478085259960853588798661929, 7.01263524643628035374465369580, 7.49505223047397846309780062344, 8.156615574972099118087121476528, 9.702734189621540522280120003755, 10.69568400798971990594891741474

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