Properties

Label 2-72-1.1-c5-0-2
Degree $2$
Conductor $72$
Sign $1$
Analytic cond. $11.5476$
Root an. cond. $3.39818$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 74·5-s − 24·7-s − 124·11-s + 478·13-s + 1.19e3·17-s + 3.04e3·19-s − 184·23-s + 2.35e3·25-s + 3.28e3·29-s − 5.72e3·31-s − 1.77e3·35-s + 1.03e4·37-s + 8.88e3·41-s − 9.18e3·43-s − 2.36e4·47-s − 1.62e4·49-s − 1.16e4·53-s − 9.17e3·55-s − 1.68e4·59-s − 1.84e4·61-s + 3.53e4·65-s − 1.55e4·67-s + 3.19e4·71-s − 4.88e3·73-s + 2.97e3·77-s + 4.45e4·79-s − 6.73e4·83-s + ⋯
L(s)  = 1  + 1.32·5-s − 0.185·7-s − 0.308·11-s + 0.784·13-s + 1.00·17-s + 1.93·19-s − 0.0725·23-s + 0.752·25-s + 0.724·29-s − 1.07·31-s − 0.245·35-s + 1.24·37-s + 0.825·41-s − 0.757·43-s − 1.56·47-s − 0.965·49-s − 0.571·53-s − 0.409·55-s − 0.631·59-s − 0.635·61-s + 1.03·65-s − 0.422·67-s + 0.752·71-s − 0.107·73-s + 0.0572·77-s + 0.803·79-s − 1.07·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(11.5476\)
Root analytic conductor: \(3.39818\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{72} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.243152870\)
\(L(\frac12)\) \(\approx\) \(2.243152870\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good5 \( 1 - 74 T + p^{5} T^{2} \)
7 \( 1 + 24 T + p^{5} T^{2} \)
11 \( 1 + 124 T + p^{5} T^{2} \)
13 \( 1 - 478 T + p^{5} T^{2} \)
17 \( 1 - 1198 T + p^{5} T^{2} \)
19 \( 1 - 3044 T + p^{5} T^{2} \)
23 \( 1 + 8 p T + p^{5} T^{2} \)
29 \( 1 - 3282 T + p^{5} T^{2} \)
31 \( 1 + 5728 T + p^{5} T^{2} \)
37 \( 1 - 10326 T + p^{5} T^{2} \)
41 \( 1 - 8886 T + p^{5} T^{2} \)
43 \( 1 + 9188 T + p^{5} T^{2} \)
47 \( 1 + 23664 T + p^{5} T^{2} \)
53 \( 1 + 11686 T + p^{5} T^{2} \)
59 \( 1 + 16876 T + p^{5} T^{2} \)
61 \( 1 + 18482 T + p^{5} T^{2} \)
67 \( 1 + 15532 T + p^{5} T^{2} \)
71 \( 1 - 31960 T + p^{5} T^{2} \)
73 \( 1 + 4886 T + p^{5} T^{2} \)
79 \( 1 - 44560 T + p^{5} T^{2} \)
83 \( 1 + 67364 T + p^{5} T^{2} \)
89 \( 1 + 71994 T + p^{5} T^{2} \)
97 \( 1 - 48866 T + p^{5} T^{2} \)
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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

−13.68737349907024953019653242729, −12.76964644452783077338102259658, −11.38043042925251573187395982764, −10.05506695862884531338147493620, −9.356379515520408623778134266069, −7.79570428857241830029409928803, −6.23683204360376539624398968655, −5.25678906530339477549974333498, −3.13211243653450481154518045274, −1.36135181700677012987976038240, 1.36135181700677012987976038240, 3.13211243653450481154518045274, 5.25678906530339477549974333498, 6.23683204360376539624398968655, 7.79570428857241830029409928803, 9.356379515520408623778134266069, 10.05506695862884531338147493620, 11.38043042925251573187395982764, 12.76964644452783077338102259658, 13.68737349907024953019653242729

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