Properties

Label 2-7360-1.1-c1-0-111
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 − 2·9-s + 2·11-s + 5·13-s + 15-s − 4·17-s − 2·19-s + 23-s + 25-s + 5·27-s + 3·29-s − 7·31-s − 2·33-s + 2·37-s − 5·39-s − 9·41-s − 4·43-s + 2·45-s + 9·47-s − 7·49-s + 4·51-s + 6·53-s − 2·55-s + 2·57-s − 2·61-s − 5·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.603·11-s + 1.38·13-s + 0.258·15-s − 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 1.25·31-s − 0.348·33-s + 0.328·37-s − 0.800·39-s − 1.40·41-s − 0.609·43-s + 0.298·45-s + 1.31·47-s − 49-s + 0.560·51-s + 0.824·53-s − 0.269·55-s + 0.264·57-s − 0.256·61-s − 0.620·65-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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.51282980127230844041269754162, −6.57259955618011446432354032044, −6.35767815900295924967002307264, −5.48141761704551174444122937443, −4.75448749320085702861476856611, −3.90202612496681922874080243166, −3.32798042816200535822847402720, −2.20575522053657701332358872144, −1.12145775198436889524504268522, 0, 1.12145775198436889524504268522, 2.20575522053657701332358872144, 3.32798042816200535822847402720, 3.90202612496681922874080243166, 4.75448749320085702861476856611, 5.48141761704551174444122937443, 6.35767815900295924967002307264, 6.57259955618011446432354032044, 7.51282980127230844041269754162

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