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Properties

Label 2-637-13.4-c1-0-13
Degree $2$
Conductor $637$
Sign $0.511 + 0.859i$
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)4-s − 3.60i·5-s + 1.73i·8-s + (1.5 + 2.59i)9-s + (−3.12 + 5.40i)10-s + (3 + 1.73i)11-s + (3.12 + 1.80i)13-s + (2.49 − 4.33i)16-s + (3.12 + 5.40i)17-s − 5.19i·18-s + (6.24 − 3.60i)19-s + (3.12 − 1.80i)20-s + (−3 − 5.19i)22-s + (−2 + 3.46i)23-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (0.250 + 0.433i)4-s − 1.61i·5-s + 0.612i·8-s + (0.5 + 0.866i)9-s + (−0.987 + 1.71i)10-s + (0.904 + 0.522i)11-s + (0.866 + 0.499i)13-s + (0.624 − 1.08i)16-s + (0.757 + 1.31i)17-s − 1.22i·18-s + (1.43 − 0.827i)19-s + (0.698 − 0.403i)20-s + (−0.639 − 1.10i)22-s + (−0.417 + 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.511 + 0.859i$
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: \((2,\ 637,\ (\ :1/2),\ 0.511 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837406 - 0.476344i\)
\(L(\frac12)\) \(\approx\) \(0.837406 - 0.476344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 - 1.80i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.24 + 3.60i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (1.5 + 0.866i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.12 + 1.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.21iT - 47T^{2} \)
53 \( 1 - 5T + 53T^{2} \)
59 \( 1 + (-6.24 + 3.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.12 + 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12 - 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 7.21iT - 83T^{2} \)
89 \( 1 + (6.24 + 3.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.24 + 3.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.16499447752018331100702999557, −9.544833005724576413440401206660, −8.824323120490776585586525113889, −8.212496328520291359051336539215, −7.28794309194263491909331314502, −5.70310504524885945680930765128, −4.89692257069560783077233991617, −3.80574935754804569119346087586, −1.75789193504972211826357711751, −1.19814982068977736533655416662, 1.02563077160050588388763888264, 3.19363821162491801652653090674, 3.74091289655486929302239605029, 5.81097095487179657548977896476, 6.59621244688971070378589983007, 7.17662170439854535957314117327, 7.986663527885752340171478667088, 9.012566800554134632786260172048, 9.906197617094972028072972426981, 10.26612761494226351619666636903

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