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Properties

Label 2-637-13.12-c1-0-37
Degree $2$
Conductor $637$
Sign $i$
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  + 2·4-s − 3.60i·5-s − 3·9-s − 3.60i·13-s + 4·16-s − 3.60i·19-s − 7.21i·20-s + 23-s − 7.99·25-s − 5·29-s + 10.8i·31-s − 6·36-s − 7.21i·41-s + 9·43-s + 10.8i·45-s + ⋯
L(s)  = 1  + 4-s − 1.61i·5-s − 9-s − 0.999i·13-s + 16-s − 0.827i·19-s − 1.61i·20-s + 0.208·23-s − 1.59·25-s − 0.928·29-s + 1.94i·31-s − 36-s − 1.12i·41-s + 1.37·43-s + 1.61i·45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15548 - 1.15548i\)
\(L(\frac12)\) \(\approx\) \(1.15548 - 1.15548i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.60iT \)
good2 \( 1 - 2T^{2} \)
3 \( 1 + 3T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.60iT - 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 10.8iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + 3.60iT - 47T^{2} \)
53 \( 1 - 11T + 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 - 18.0iT - 83T^{2} \)
89 \( 1 + 3.60iT - 89T^{2} \)
97 \( 1 - 18.0iT - 97T^{2} \)
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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

−10.53758706961148283263872825775, −9.310666724135831601117130001293, −8.599949348034413221895450657824, −7.87888024039073588277570836718, −6.80306429687961310874924384934, −5.54240797548505922390033308921, −5.18648095530453650072178156842, −3.60556027641073429976861898300, −2.36122142279252356623599827622, −0.865643589004246538352849698417, 2.11003093269110094512873523839, 2.90183652915865826561172861350, 3.93798235111331440903582746562, 5.85277215831893289814021448477, 6.24538692255915754878503141910, 7.26779433921307144355118806726, 7.82472286374025567874595526250, 9.177948290762569783891091012030, 10.17301053760386419033308267863, 10.95501419740165719230398927607

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