Properties

Label 2-7360-1.1-c1-0-144
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.264·3-s − 5-s + 2.69·7-s − 2.93·9-s − 0.153·11-s + 3.71·13-s − 0.264·15-s − 1.55·17-s + 2.13·19-s + 0.712·21-s + 23-s + 25-s − 1.56·27-s − 4.73·29-s − 4.53·31-s − 0.0405·33-s − 2.69·35-s − 5.51·37-s + 0.982·39-s − 11.0·41-s + 6.27·43-s + 2.93·45-s − 12.5·47-s + 0.284·49-s − 0.409·51-s − 8.58·53-s + 0.153·55-s + ⋯
L(s)  = 1  + 0.152·3-s − 0.447·5-s + 1.02·7-s − 0.976·9-s − 0.0463·11-s + 1.03·13-s − 0.0681·15-s − 0.376·17-s + 0.489·19-s + 0.155·21-s + 0.208·23-s + 0.200·25-s − 0.301·27-s − 0.878·29-s − 0.814·31-s − 0.00706·33-s − 0.456·35-s − 0.907·37-s + 0.157·39-s − 1.72·41-s + 0.956·43-s + 0.436·45-s − 1.83·47-s + 0.0406·49-s − 0.0573·51-s − 1.17·53-s + 0.0207·55-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: \(1\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.264T + 3T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 + 0.153T + 11T^{2} \)
13 \( 1 - 3.71T + 13T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 4.53T + 31T^{2} \)
37 \( 1 + 5.51T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 6.27T + 43T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + 8.58T + 53T^{2} \)
59 \( 1 + 0.242T + 59T^{2} \)
61 \( 1 - 7.24T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 6.89T + 73T^{2} \)
79 \( 1 - 8.28T + 79T^{2} \)
83 \( 1 - 3.40T + 83T^{2} \)
89 \( 1 - 0.953T + 89T^{2} \)
97 \( 1 - 6.49T + 97T^{2} \)
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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

−7.79232266395328642199442142035, −6.88907934433778249302077867084, −6.17090689926030102858643910163, −5.27170800342165458121915143928, −4.90480892059887718603265545792, −3.73285467489948719403207297432, −3.34755295182695339728857760786, −2.17925367745728844417998810416, −1.36092079391319389919989558991, 0, 1.36092079391319389919989558991, 2.17925367745728844417998810416, 3.34755295182695339728857760786, 3.73285467489948719403207297432, 4.90480892059887718603265545792, 5.27170800342165458121915143928, 6.17090689926030102858643910163, 6.88907934433778249302077867084, 7.79232266395328642199442142035

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