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Properties

Label 2-369600-1.1-c1-0-480
Degree $2$
Conductor $369600$
Sign $-1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 + 7-s + 9-s + 11-s − 2·13-s − 2·17-s + 4·19-s + 21-s + 27-s + 2·29-s + 8·31-s + 33-s − 10·37-s − 2·39-s − 6·41-s + 4·43-s + 49-s − 2·51-s − 2·53-s + 4·57-s − 12·59-s − 14·61-s + 63-s − 12·67-s + 6·73-s + 77-s + 16·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s − 1.56·59-s − 1.79·61-s + 0.125·63-s − 1.46·67-s + 0.702·73-s + 0.113·77-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.53472453836239, −12.31206571069081, −11.89320803508718, −11.47060925178867, −10.80789302945416, −10.48478593594242, −9.971380343343296, −9.527907871714639, −9.041438731012802, −8.695476753794948, −8.151906895238801, −7.718373482940025, −7.270576346592727, −6.840287068199385, −6.251523545637473, −5.838124793859368, −5.026699579259159, −4.729004103247498, −4.373703186190853, −3.498136913868233, −3.226675493504383, −2.642190300736994, −1.985096388887825, −1.520396408179228, −0.8537293204330090, 0, 0.8537293204330090, 1.520396408179228, 1.985096388887825, 2.642190300736994, 3.226675493504383, 3.498136913868233, 4.373703186190853, 4.729004103247498, 5.026699579259159, 5.838124793859368, 6.251523545637473, 6.840287068199385, 7.270576346592727, 7.718373482940025, 8.151906895238801, 8.695476753794948, 9.041438731012802, 9.527907871714639, 9.971380343343296, 10.48478593594242, 10.80789302945416, 11.47060925178867, 11.89320803508718, 12.31206571069081, 12.53472453836239

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