Properties

Label 2-9360-1.1-c1-0-32
Degree $2$
Conductor $9360$
Sign $1$
Analytic cond. $74.7399$
Root an. cond. $8.64522$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s + 4·11-s − 13-s − 8·17-s + 6·19-s + 6·23-s + 25-s + 4·29-s − 2·35-s − 2·37-s + 2·41-s + 4·43-s − 3·49-s + 10·53-s + 4·55-s + 4·59-s − 10·61-s − 65-s − 12·67-s − 8·71-s − 8·73-s − 8·77-s − 8·79-s + 12·83-s − 8·85-s + 14·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.277·13-s − 1.94·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s + 1.37·53-s + 0.539·55-s + 0.520·59-s − 1.28·61-s − 0.124·65-s − 1.46·67-s − 0.949·71-s − 0.936·73-s − 0.911·77-s − 0.900·79-s + 1.31·83-s − 0.867·85-s + 1.48·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9360\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(74.7399\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.132313219\)
\(L(\frac12)\) \(\approx\) \(2.132313219\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.44402085815186928221316548841, −6.97292458083943566285937572696, −6.40072740812642782744428493784, −5.80778030856188485702098155254, −4.84659241124141227396461053825, −4.29896194556588441618526638864, −3.32325696980069181275040043023, −2.71173310432285834822225949996, −1.70630519538430677611499253161, −0.71285530202580182453173395478, 0.71285530202580182453173395478, 1.70630519538430677611499253161, 2.71173310432285834822225949996, 3.32325696980069181275040043023, 4.29896194556588441618526638864, 4.84659241124141227396461053825, 5.80778030856188485702098155254, 6.40072740812642782744428493784, 6.97292458083943566285937572696, 7.44402085815186928221316548841

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