Properties

Label 2-637-1.1-c1-0-6
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  − 2.44·2-s − 0.667·3-s + 3.99·4-s + 0.910·5-s + 1.63·6-s − 4.87·8-s − 2.55·9-s − 2.22·10-s + 3.67·11-s − 2.66·12-s + 13-s − 0.607·15-s + 3.95·16-s + 7.18·17-s + 6.25·18-s − 1.97·19-s + 3.63·20-s − 9.00·22-s − 0.596·23-s + 3.25·24-s − 4.17·25-s − 2.44·26-s + 3.70·27-s − 3.64·29-s + 1.48·30-s − 7.08·31-s + 0.0786·32-s + ⋯
L(s)  = 1  − 1.73·2-s − 0.385·3-s + 1.99·4-s + 0.407·5-s + 0.667·6-s − 1.72·8-s − 0.851·9-s − 0.704·10-s + 1.10·11-s − 0.769·12-s + 0.277·13-s − 0.156·15-s + 0.987·16-s + 1.74·17-s + 1.47·18-s − 0.453·19-s + 0.812·20-s − 1.91·22-s − 0.124·23-s + 0.664·24-s − 0.834·25-s − 0.480·26-s + 0.713·27-s − 0.677·29-s + 0.271·30-s − 1.27·31-s + 0.0139·32-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

\(L(1)\) \(\approx\) \(0.6122053027\)
\(L(\frac12)\) \(\approx\) \(0.6122053027\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 2T^{2} \)
3 \( 1 + 0.667T + 3T^{2} \)
5 \( 1 - 0.910T + 5T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 + 0.596T + 23T^{2} \)
29 \( 1 + 3.64T + 29T^{2} \)
31 \( 1 + 7.08T + 31T^{2} \)
37 \( 1 - 0.710T + 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 9.58T + 59T^{2} \)
61 \( 1 - 6.98T + 61T^{2} \)
67 \( 1 - 1.22T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 6.53T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 7.16T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 9.09T + 97T^{2} \)
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

−10.46466947777893245514847469291, −9.488190567588973886924029949260, −9.086066172778489900924812522211, −8.077294373094889802145067715754, −7.32571527757451021684283141226, −6.20714984492076393067546545162, −5.61686331131908083242876772755, −3.70465499603267629391003055171, −2.19542371399468063645972123031, −0.907931743152917920890736717421, 0.907931743152917920890736717421, 2.19542371399468063645972123031, 3.70465499603267629391003055171, 5.61686331131908083242876772755, 6.20714984492076393067546545162, 7.32571527757451021684283141226, 8.077294373094889802145067715754, 9.086066172778489900924812522211, 9.488190567588973886924029949260, 10.46466947777893245514847469291

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