Properties

Label 2-76440-1.1-c1-0-4
Degree $2$
Conductor $76440$
Sign $1$
Analytic cond. $610.376$
Root an. cond. $24.7057$
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  + 3-s − 5-s + 9-s + 4·11-s − 13-s − 15-s − 6·17-s − 4·23-s + 25-s + 27-s − 6·29-s + 8·31-s + 4·33-s − 2·37-s − 39-s − 10·41-s − 4·43-s − 45-s − 8·47-s − 6·51-s − 2·53-s − 4·55-s − 4·59-s − 14·61-s + 65-s − 12·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 1.45·17-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 0.840·51-s − 0.274·53-s − 0.539·55-s − 0.520·59-s − 1.79·61-s + 0.124·65-s − 1.46·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76440\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(610.376\)
Root analytic conductor: \(24.7057\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.511490216\)
\(L(\frac12)\) \(\approx\) \(1.511490216\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.94173770571009, −13.59281626163068, −13.24061496009055, −12.45839438126691, −12.00604075571725, −11.67688031074574, −11.07493401093223, −10.57570600863106, −9.836908841235221, −9.545660718199590, −8.841683994189123, −8.575105630048673, −7.971415918875595, −7.430119322656019, −6.750247476248943, −6.491102495508007, −5.849535736862680, −4.887015439426219, −4.487923688088494, −3.991114890852368, −3.330474093247808, −2.832410062996245, −1.809188269481281, −1.625382000011567, −0.3699734702662848, 0.3699734702662848, 1.625382000011567, 1.809188269481281, 2.832410062996245, 3.330474093247808, 3.991114890852368, 4.487923688088494, 4.887015439426219, 5.849535736862680, 6.491102495508007, 6.750247476248943, 7.430119322656019, 7.971415918875595, 8.575105630048673, 8.841683994189123, 9.545660718199590, 9.836908841235221, 10.57570600863106, 11.07493401093223, 11.67688031074574, 12.00604075571725, 12.45839438126691, 13.24061496009055, 13.59281626163068, 13.94173770571009

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