Properties

Label 2-637-1.1-c1-0-19
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  + 0.264·2-s − 2.90·3-s − 1.92·4-s + 1.43·5-s − 0.769·6-s − 1.03·8-s + 5.46·9-s + 0.379·10-s + 5.50·11-s + 5.61·12-s − 13-s − 4.17·15-s + 3.58·16-s − 4.83·17-s + 1.44·18-s − 2.82·19-s − 2.76·20-s + 1.45·22-s − 5.99·23-s + 3.02·24-s − 2.94·25-s − 0.264·26-s − 7.16·27-s + 1.04·29-s − 1.10·30-s − 9.20·31-s + 3.02·32-s + ⋯
L(s)  = 1  + 0.187·2-s − 1.67·3-s − 0.964·4-s + 0.641·5-s − 0.314·6-s − 0.367·8-s + 1.82·9-s + 0.120·10-s + 1.65·11-s + 1.62·12-s − 0.277·13-s − 1.07·15-s + 0.896·16-s − 1.17·17-s + 0.340·18-s − 0.647·19-s − 0.619·20-s + 0.310·22-s − 1.25·23-s + 0.617·24-s − 0.588·25-s − 0.0518·26-s − 1.37·27-s + 0.193·29-s − 0.201·30-s − 1.65·31-s + 0.535·32-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 \( 1 - 0.264T + 2T^{2} \)
3 \( 1 + 2.90T + 3T^{2} \)
5 \( 1 - 1.43T + 5T^{2} \)
11 \( 1 - 5.50T + 11T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 5.99T + 23T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + 9.20T + 31T^{2} \)
37 \( 1 - 0.612T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 8.43T + 43T^{2} \)
47 \( 1 + 2.40T + 47T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 - 0.870T + 59T^{2} \)
61 \( 1 - 3.33T + 61T^{2} \)
67 \( 1 + 6.62T + 67T^{2} \)
71 \( 1 + 6.85T + 71T^{2} \)
73 \( 1 + 3.14T + 73T^{2} \)
79 \( 1 + 17.5T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 0.995T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−10.12584741197304932989483512538, −9.484951424008642097738591162571, −8.643777902779332689074712841360, −7.12234778341550234254355027653, −6.16326475512675874295743930417, −5.75795280344623294046556809373, −4.57038187557279817392660739404, −4.00195362636389187768498287889, −1.64873412580056810232984917262, 0, 1.64873412580056810232984917262, 4.00195362636389187768498287889, 4.57038187557279817392660739404, 5.75795280344623294046556809373, 6.16326475512675874295743930417, 7.12234778341550234254355027653, 8.643777902779332689074712841360, 9.484951424008642097738591162571, 10.12584741197304932989483512538

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