Properties

Label 2-7360-1.1-c1-0-129
Degree $2$
Conductor $7360$
Sign $-1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·7-s − 2·9-s + 4·11-s − 13-s − 15-s + 4·19-s − 2·21-s − 23-s + 25-s − 5·27-s + 7·29-s − 7·31-s + 4·33-s + 2·35-s + 4·37-s − 39-s + 3·41-s − 6·43-s + 2·45-s − 13·47-s − 3·49-s − 10·53-s − 4·55-s + 4·57-s + 8·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.29·29-s − 1.25·31-s + 0.696·33-s + 0.338·35-s + 0.657·37-s − 0.160·39-s + 0.468·41-s − 0.914·43-s + 0.298·45-s − 1.89·47-s − 3/7·49-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 1.04·59-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: yes
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 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.71063408375395939308821748960, −6.73102080244819017386355921436, −6.40935477229377881403943792274, −5.44143623003953362474001114493, −4.64700599053745727690123410905, −3.61383197772591189051377594869, −3.33469446734602157165360726382, −2.42388957128345938101894065388, −1.28108361829890287732195514879, 0, 1.28108361829890287732195514879, 2.42388957128345938101894065388, 3.33469446734602157165360726382, 3.61383197772591189051377594869, 4.64700599053745727690123410905, 5.44143623003953362474001114493, 6.40935477229377881403943792274, 6.73102080244819017386355921436, 7.71063408375395939308821748960

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