Properties

Label 2-346560-1.1-c1-0-165
Degree $2$
Conductor $346560$
Sign $-1$
Analytic cond. $2767.29$
Root an. cond. $52.6050$
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 + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 4·23-s + 25-s − 27-s + 6·29-s − 4·31-s + 4·33-s − 6·37-s − 2·39-s − 10·41-s + 4·43-s + 45-s − 12·47-s − 7·49-s − 2·51-s + 6·53-s − 4·55-s − 12·59-s + 2·61-s + 2·65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(346560\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(2767.29\)
Root analytic conductor: \(52.6050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{346560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 346560,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
19 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−12.77711479706090, −12.35093510166900, −11.91785995287093, −11.36572639920434, −10.86773184493215, −10.59564035219110, −10.12455219307558, −9.744224396513745, −9.195520120460597, −8.583632509386848, −8.253965519580463, −7.739519148444873, −7.145818707990325, −6.713786064659241, −6.308272430239916, −5.612787835380646, −5.392637599407813, −4.820885586410729, −4.510660997994715, −3.573769599834181, −3.178550079761797, −2.714963351666588, −1.831833484838075, −1.509394035386220, −0.6939207875199140, 0, 0.6939207875199140, 1.509394035386220, 1.831833484838075, 2.714963351666588, 3.178550079761797, 3.573769599834181, 4.510660997994715, 4.820885586410729, 5.392637599407813, 5.612787835380646, 6.308272430239916, 6.713786064659241, 7.145818707990325, 7.739519148444873, 8.253965519580463, 8.583632509386848, 9.195520120460597, 9.744224396513745, 10.12455219307558, 10.59564035219110, 10.86773184493215, 11.36572639920434, 11.91785995287093, 12.35093510166900, 12.77711479706090

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