Properties

Label 2-637-1.1-c1-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.26·2-s − 2.62·3-s − 0.398·4-s − 2.90·5-s − 3.32·6-s − 3.03·8-s + 3.90·9-s − 3.67·10-s + 2.03·11-s + 1.04·12-s + 13-s + 7.62·15-s − 3.04·16-s + 3.99·17-s + 4.93·18-s + 6.96·19-s + 1.15·20-s + 2.57·22-s − 0.627·23-s + 7.97·24-s + 3.42·25-s + 1.26·26-s − 2.37·27-s + 1.09·29-s + 9.65·30-s − 10.4·31-s + 2.21·32-s + ⋯
L(s)  = 1  + 0.894·2-s − 1.51·3-s − 0.199·4-s − 1.29·5-s − 1.35·6-s − 1.07·8-s + 1.30·9-s − 1.16·10-s + 0.614·11-s + 0.302·12-s + 0.277·13-s + 1.96·15-s − 0.761·16-s + 0.969·17-s + 1.16·18-s + 1.59·19-s + 0.258·20-s + 0.549·22-s − 0.130·23-s + 1.62·24-s + 0.685·25-s + 0.248·26-s − 0.456·27-s + 0.203·29-s + 1.76·30-s − 1.87·31-s + 0.391·32-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

\(L(1)\) \(\approx\) \(0.8488849415\)
\(L(\frac12)\) \(\approx\) \(0.8488849415\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.26T + 2T^{2} \)
3 \( 1 + 2.62T + 3T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
17 \( 1 - 3.99T + 17T^{2} \)
19 \( 1 - 6.96T + 19T^{2} \)
23 \( 1 + 0.627T + 23T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 + 0.521T + 41T^{2} \)
43 \( 1 - 0.329T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 7.11T + 53T^{2} \)
59 \( 1 - 2.03T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 - 2.97T + 73T^{2} \)
79 \( 1 + 8.76T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + 2.68T + 89T^{2} \)
97 \( 1 + 2.32T + 97T^{2} \)
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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

−11.04710187344569022810355330165, −9.903102968238379156622246185888, −8.870415460217353437468124255515, −7.64760966936838998623177031553, −6.82347048110939850676750561411, −5.65871437213022781620982719046, −5.22160992529478884201996075507, −4.07052869479064403728805460215, −3.46232434429809137797318091278, −0.75742506270143291825187926934, 0.75742506270143291825187926934, 3.46232434429809137797318091278, 4.07052869479064403728805460215, 5.22160992529478884201996075507, 5.65871437213022781620982719046, 6.82347048110939850676750561411, 7.64760966936838998623177031553, 8.870415460217353437468124255515, 9.903102968238379156622246185888, 11.04710187344569022810355330165

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