Properties

Label 2-7360-1.1-c1-0-115
Degree $2$
Conductor $7360$
Sign $1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
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  + 3.30·3-s + 5-s + 2.55·7-s + 7.93·9-s − 2.72·11-s − 7.12·13-s + 3.30·15-s + 0.924·17-s + 7.51·19-s + 8.43·21-s + 23-s + 25-s + 16.3·27-s + 2.38·29-s − 0.866·31-s − 9.00·33-s + 2.55·35-s − 0.352·37-s − 23.5·39-s + 4.34·41-s + 7.93·45-s + 13.3·47-s − 0.495·49-s + 3.05·51-s − 3.99·53-s − 2.72·55-s + 24.8·57-s + ⋯
L(s)  = 1  + 1.90·3-s + 0.447·5-s + 0.963·7-s + 2.64·9-s − 0.821·11-s − 1.97·13-s + 0.853·15-s + 0.224·17-s + 1.72·19-s + 1.84·21-s + 0.208·23-s + 0.200·25-s + 3.13·27-s + 0.442·29-s − 0.155·31-s − 1.56·33-s + 0.431·35-s − 0.0580·37-s − 3.77·39-s + 0.677·41-s + 1.18·45-s + 1.94·47-s − 0.0707·49-s + 0.427·51-s − 0.548·53-s − 0.367·55-s + 3.29·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7360\)    =    \(2^{6} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(58.7698\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.318723056\)
\(L(\frac12)\) \(\approx\) \(5.318723056\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 - 3.30T + 3T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 + 7.12T + 13T^{2} \)
17 \( 1 - 0.924T + 17T^{2} \)
19 \( 1 - 7.51T + 19T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 + 0.866T + 31T^{2} \)
37 \( 1 + 0.352T + 37T^{2} \)
41 \( 1 - 4.34T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 + 3.99T + 53T^{2} \)
59 \( 1 + 3.84T + 59T^{2} \)
61 \( 1 - 9.14T + 61T^{2} \)
67 \( 1 + 3.15T + 67T^{2} \)
71 \( 1 - 6.07T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 6.35T + 83T^{2} \)
89 \( 1 + 9.71T + 89T^{2} \)
97 \( 1 - 8.76T + 97T^{2} \)
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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

−7.85927568624737851784262018763, −7.43112540206226827572324827308, −7.03362488203158340459035299054, −5.55821206125220765393065336215, −4.93683901474265240348888881863, −4.34625666258927739442047764911, −3.24534008473767542906689724277, −2.62769496148924198947916521796, −2.12587449108878971024711813678, −1.13023790815859447435064246334, 1.13023790815859447435064246334, 2.12587449108878971024711813678, 2.62769496148924198947916521796, 3.24534008473767542906689724277, 4.34625666258927739442047764911, 4.93683901474265240348888881863, 5.55821206125220765393065336215, 7.03362488203158340459035299054, 7.43112540206226827572324827308, 7.85927568624737851784262018763

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