Properties

Label 2-637-1.1-c1-0-21
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 $1$

Origins

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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: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 9.09T + 97T^{2} \)
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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

−9.748705149106110196372907742058, −9.252775584778306557439180299938, −8.496596284383703349614529315625, −7.85830550016830784478718439741, −6.88285217326742670782616051395, −6.10222335697885023933537377020, −4.36426167714966574176005306187, −2.94579331898935834227519856446, −1.74677162868966491552739570492, 0, 1.74677162868966491552739570492, 2.94579331898935834227519856446, 4.36426167714966574176005306187, 6.10222335697885023933537377020, 6.88285217326742670782616051395, 7.85830550016830784478718439741, 8.496596284383703349614529315625, 9.252775584778306557439180299938, 9.748705149106110196372907742058

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