Properties

Label 2-1232-1.1-c3-0-27
Degree $2$
Conductor $1232$
Sign $1$
Analytic cond. $72.6903$
Root an. cond. $8.52586$
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  + 2·3-s + 6.48·5-s − 7·7-s − 23·9-s + 11·11-s − 45.6·13-s + 12.9·15-s − 63.6·17-s + 39.5·19-s − 14·21-s + 78.9·23-s − 82.9·25-s − 100·27-s + 256.·29-s + 170.·31-s + 22·33-s − 45.3·35-s − 223.·37-s − 91.3·39-s + 307.·41-s + 316·43-s − 149.·45-s + 576.·47-s + 49·49-s − 127.·51-s + 173.·53-s + 71.3·55-s + ⋯
L(s)  = 1  + 0.384·3-s + 0.580·5-s − 0.377·7-s − 0.851·9-s + 0.301·11-s − 0.974·13-s + 0.223·15-s − 0.908·17-s + 0.477·19-s − 0.145·21-s + 0.715·23-s − 0.663·25-s − 0.712·27-s + 1.64·29-s + 0.989·31-s + 0.116·33-s − 0.219·35-s − 0.995·37-s − 0.374·39-s + 1.17·41-s + 1.12·43-s − 0.494·45-s + 1.78·47-s + 0.142·49-s − 0.349·51-s + 0.450·53-s + 0.174·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(72.6903\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.186928386\)
\(L(\frac12)\) \(\approx\) \(2.186928386\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 7T \)
11 \( 1 - 11T \)
good3 \( 1 - 2T + 27T^{2} \)
5 \( 1 - 6.48T + 125T^{2} \)
13 \( 1 + 45.6T + 2.19e3T^{2} \)
17 \( 1 + 63.6T + 4.91e3T^{2} \)
19 \( 1 - 39.5T + 6.85e3T^{2} \)
23 \( 1 - 78.9T + 1.21e4T^{2} \)
29 \( 1 - 256.T + 2.43e4T^{2} \)
31 \( 1 - 170.T + 2.97e4T^{2} \)
37 \( 1 + 223.T + 5.06e4T^{2} \)
41 \( 1 - 307.T + 6.89e4T^{2} \)
43 \( 1 - 316T + 7.95e4T^{2} \)
47 \( 1 - 576.T + 1.03e5T^{2} \)
53 \( 1 - 173.T + 1.48e5T^{2} \)
59 \( 1 - 82.6T + 2.05e5T^{2} \)
61 \( 1 + 63.1T + 2.26e5T^{2} \)
67 \( 1 - 349.T + 3.00e5T^{2} \)
71 \( 1 - 119.T + 3.57e5T^{2} \)
73 \( 1 - 573.T + 3.89e5T^{2} \)
79 \( 1 + 568.T + 4.93e5T^{2} \)
83 \( 1 + 510.T + 5.71e5T^{2} \)
89 \( 1 - 1.09e3T + 7.04e5T^{2} \)
97 \( 1 - 396.T + 9.12e5T^{2} \)
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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

−9.246275270585004182215590171294, −8.736401662797506669992087225185, −7.72725709705183425753292356845, −6.83118906970950923231376472259, −6.02593403694438886544611789274, −5.15574739228956851092072636333, −4.12588862236983161099526375972, −2.86294757405288492584291718970, −2.29681166545386736588338673051, −0.71968827396840271895515038922, 0.71968827396840271895515038922, 2.29681166545386736588338673051, 2.86294757405288492584291718970, 4.12588862236983161099526375972, 5.15574739228956851092072636333, 6.02593403694438886544611789274, 6.83118906970950923231376472259, 7.72725709705183425753292356845, 8.736401662797506669992087225185, 9.246275270585004182215590171294

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