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Properties

Label 2-369600-1.1-c1-0-332
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 − 6·29-s + 33-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s + 49-s − 2·51-s − 2·53-s − 4·57-s − 4·59-s − 6·61-s − 63-s + 12·67-s − 10·73-s − 77-s + 8·79-s + 81-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 − 1.11·29-s + 0.174·33-s + 0.986·37-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 − 0.520·59-s − 0.768·61-s − 0.125·63-s + 1.46·67-s − 1.17·73-s − 0.113·77-s + 0.900·79-s + 1/9·81-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 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.83598337123329, −12.31285391604760, −11.90807702423702, −11.28449467428022, −10.99726182908473, −10.35685519298846, −9.981419876854397, −9.520867050733714, −9.041500889907260, −8.785712244792687, −8.065515827728517, −7.823326305884723, −7.199286086694448, −6.707455697428894, −6.403044708183355, −5.775979277277442, −5.236685516461251, −4.589686109321416, −4.248416561619140, −3.656746838707716, −3.161909833750696, −2.615114812608571, −1.986358059073187, −1.651118726190154, −0.6818563128175179, 0, 0.6818563128175179, 1.651118726190154, 1.986358059073187, 2.615114812608571, 3.161909833750696, 3.656746838707716, 4.248416561619140, 4.589686109321416, 5.236685516461251, 5.775979277277442, 6.403044708183355, 6.707455697428894, 7.199286086694448, 7.823326305884723, 8.065515827728517, 8.785712244792687, 9.041500889907260, 9.520867050733714, 9.981419876854397, 10.35685519298846, 10.99726182908473, 11.28449467428022, 11.90807702423702, 12.31285391604760, 12.83598337123329

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