Properties

Label 2-7360-1.1-c1-0-55
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  − 2.56·3-s − 5-s + 5.12·7-s + 3.56·9-s + 4·11-s + 0.561·13-s + 2.56·15-s − 3.12·17-s − 4·19-s − 13.1·21-s + 23-s + 25-s − 1.43·27-s + 8.56·29-s + 1.43·31-s − 10.2·33-s − 5.12·35-s + 7.12·37-s − 1.43·39-s + 0.561·41-s + 9.12·43-s − 3.56·45-s − 3.68·47-s + 19.2·49-s + 8·51-s + 4.24·53-s − 4·55-s + ⋯
L(s)  = 1  − 1.47·3-s − 0.447·5-s + 1.93·7-s + 1.18·9-s + 1.20·11-s + 0.155·13-s + 0.661·15-s − 0.757·17-s − 0.917·19-s − 2.86·21-s + 0.208·23-s + 0.200·25-s − 0.276·27-s + 1.58·29-s + 0.258·31-s − 1.78·33-s − 0.865·35-s + 1.17·37-s − 0.230·39-s + 0.0876·41-s + 1.39·43-s − 0.530·45-s − 0.537·47-s + 2.74·49-s + 1.12·51-s + 0.583·53-s − 0.539·55-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: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.576118839\)
\(L(\frac12)\) \(\approx\) \(1.576118839\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 - 8.56T + 29T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 + 3.68T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 6.24T + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−7.890069999890656493233429596824, −7.05551878643065207026123010597, −6.44216589333450312829571480362, −5.84544503805564650869391913762, −4.95437338124771029084920299580, −4.44836442739352586412918836602, −4.08780815415882570868563975380, −2.50556784308839949635437618071, −1.43911241914330338418995079514, −0.77106967266729678983673732227, 0.77106967266729678983673732227, 1.43911241914330338418995079514, 2.50556784308839949635437618071, 4.08780815415882570868563975380, 4.44836442739352586412918836602, 4.95437338124771029084920299580, 5.84544503805564650869391913762, 6.44216589333450312829571480362, 7.05551878643065207026123010597, 7.890069999890656493233429596824

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