Properties

Label 2-624-1.1-c3-0-22
Degree $2$
Conductor $624$
Sign $1$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 13.6·5-s + 15.6·7-s + 9·9-s + 50.5·11-s + 13·13-s + 40.8·15-s + 2·17-s − 18.1·19-s + 46.8·21-s + 64·23-s + 60.7·25-s + 27·27-s − 103.·29-s − 34.4·31-s + 151.·33-s + 213.·35-s − 267.·37-s + 39·39-s − 147.·41-s + 166.·43-s + 122.·45-s + 325.·47-s − 98.6·49-s + 6·51-s − 111.·53-s + 688.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.21·5-s + 0.843·7-s + 0.333·9-s + 1.38·11-s + 0.277·13-s + 0.703·15-s + 0.0285·17-s − 0.219·19-s + 0.487·21-s + 0.580·23-s + 0.486·25-s + 0.192·27-s − 0.664·29-s − 0.199·31-s + 0.799·33-s + 1.02·35-s − 1.18·37-s + 0.160·39-s − 0.561·41-s + 0.592·43-s + 0.406·45-s + 1.01·47-s − 0.287·49-s + 0.0164·51-s − 0.287·53-s + 1.68·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.847563012\)
\(L(\frac12)\) \(\approx\) \(3.847563012\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3T \)
13 \( 1 - 13T \)
good5 \( 1 - 13.6T + 125T^{2} \)
7 \( 1 - 15.6T + 343T^{2} \)
11 \( 1 - 50.5T + 1.33e3T^{2} \)
17 \( 1 - 2T + 4.91e3T^{2} \)
19 \( 1 + 18.1T + 6.85e3T^{2} \)
23 \( 1 - 64T + 1.21e4T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 + 34.4T + 2.97e4T^{2} \)
37 \( 1 + 267.T + 5.06e4T^{2} \)
41 \( 1 + 147.T + 6.89e4T^{2} \)
43 \( 1 - 166.T + 7.95e4T^{2} \)
47 \( 1 - 325.T + 1.03e5T^{2} \)
53 \( 1 + 111.T + 1.48e5T^{2} \)
59 \( 1 + 24.3T + 2.05e5T^{2} \)
61 \( 1 + 640.T + 2.26e5T^{2} \)
67 \( 1 - 382.T + 3.00e5T^{2} \)
71 \( 1 + 510.T + 3.57e5T^{2} \)
73 \( 1 + 291.T + 3.89e5T^{2} \)
79 \( 1 - 794.T + 4.93e5T^{2} \)
83 \( 1 + 945.T + 5.71e5T^{2} \)
89 \( 1 - 317.T + 7.04e5T^{2} \)
97 \( 1 - 1.57e3T + 9.12e5T^{2} \)
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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

−10.09761762605332434325181662931, −9.151614553780722590258903504894, −8.776302498750950502998872783074, −7.57218922446952078522760984649, −6.59817340332287382715892581792, −5.70370450698841604630706746641, −4.61936163324185044302251140777, −3.48484020754733292211297457744, −2.06223372633070112311336423289, −1.32977305060417499735762231754, 1.32977305060417499735762231754, 2.06223372633070112311336423289, 3.48484020754733292211297457744, 4.61936163324185044302251140777, 5.70370450698841604630706746641, 6.59817340332287382715892581792, 7.57218922446952078522760984649, 8.776302498750950502998872783074, 9.151614553780722590258903504894, 10.09761762605332434325181662931

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