Properties

Label 2-637-1.1-c1-0-29
Degree $2$
Conductor $637$
Sign $1$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.21·2-s + 2.47·3-s + 2.89·4-s − 2.12·5-s + 5.47·6-s + 1.98·8-s + 3.12·9-s − 4.69·10-s + 4.78·11-s + 7.17·12-s + 13-s − 5.25·15-s − 1.39·16-s − 3.77·17-s + 6.91·18-s − 3.56·19-s − 6.15·20-s + 10.5·22-s + 4.47·23-s + 4.92·24-s − 0.493·25-s + 2.21·26-s + 0.303·27-s − 5.90·29-s − 11.6·30-s − 3.77·31-s − 7.06·32-s + ⋯
L(s)  = 1  + 1.56·2-s + 1.42·3-s + 1.44·4-s − 0.949·5-s + 2.23·6-s + 0.703·8-s + 1.04·9-s − 1.48·10-s + 1.44·11-s + 2.07·12-s + 0.277·13-s − 1.35·15-s − 0.348·16-s − 0.916·17-s + 1.62·18-s − 0.818·19-s − 1.37·20-s + 2.25·22-s + 0.932·23-s + 1.00·24-s − 0.0987·25-s + 0.434·26-s + 0.0584·27-s − 1.09·29-s − 2.12·30-s − 0.677·31-s − 1.24·32-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(4.553843371\)
\(L(\frac12)\) \(\approx\) \(4.553843371\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.21T + 2T^{2} \)
3 \( 1 - 2.47T + 3T^{2} \)
5 \( 1 + 2.12T + 5T^{2} \)
11 \( 1 - 4.78T + 11T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + 3.77T + 31T^{2} \)
37 \( 1 - 5.62T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + 7.10T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 4.78T + 59T^{2} \)
61 \( 1 - 3.20T + 61T^{2} \)
67 \( 1 + 2.89T + 67T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 + 5.17T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 3.66T + 89T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.17224048030397281420343719952, −9.355258434925682333703744829438, −8.886195130241436145544236273743, −7.82362230684433338237776386124, −6.94545177728682669862684893793, −6.04606178243473389147000976931, −4.49434860961139003451501846553, −3.94629995237744310438685607732, −3.25133560467073639152613808583, −2.06131719640071757778923039939, 2.06131719640071757778923039939, 3.25133560467073639152613808583, 3.94629995237744310438685607732, 4.49434860961139003451501846553, 6.04606178243473389147000976931, 6.94545177728682669862684893793, 7.82362230684433338237776386124, 8.886195130241436145544236273743, 9.355258434925682333703744829438, 11.17224048030397281420343719952

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