Properties

Label 2-63-1.1-c1-0-0
Degree $2$
Conductor $63$
Sign $1$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
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  − 1.73·2-s + 0.999·4-s + 3.46·5-s + 7-s + 1.73·8-s − 5.99·10-s − 3.46·11-s + 2·13-s − 1.73·14-s − 5·16-s − 3.46·17-s − 4·19-s + 3.46·20-s + 5.99·22-s + 3.46·23-s + 6.99·25-s − 3.46·26-s + 0.999·28-s − 4·31-s + 5.19·32-s + 5.99·34-s + 3.46·35-s + 2·37-s + 6.92·38-s + 6.00·40-s − 10.3·41-s − 4·43-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s + 1.54·5-s + 0.377·7-s + 0.612·8-s − 1.89·10-s − 1.04·11-s + 0.554·13-s − 0.462·14-s − 1.25·16-s − 0.840·17-s − 0.917·19-s + 0.774·20-s + 1.27·22-s + 0.722·23-s + 1.39·25-s − 0.679·26-s + 0.188·28-s − 0.718·31-s + 0.918·32-s + 1.02·34-s + 0.585·35-s + 0.328·37-s + 1.12·38-s + 0.948·40-s − 1.62·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5995131227\)
\(L(\frac12)\) \(\approx\) \(0.5995131227\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
good2 \( 1 + 1.73T + 2T^{2} \)
5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−15.11316081447122050209031377631, −13.65130748536723378296433633348, −13.09139824524204740103542304769, −10.98695612250120843647135121140, −10.27329396446645584897172286523, −9.194175210866369558732456620981, −8.258563185490701385396362178951, −6.66210332331203199003882891704, −5.10078856423729307758368068237, −2.00695843206272168824484431744, 2.00695843206272168824484431744, 5.10078856423729307758368068237, 6.66210332331203199003882891704, 8.258563185490701385396362178951, 9.194175210866369558732456620981, 10.27329396446645584897172286523, 10.98695612250120843647135121140, 13.09139824524204740103542304769, 13.65130748536723378296433633348, 15.11316081447122050209031377631

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