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Properties

Label 2-76230-1.1-c1-0-57
Degree $2$
Conductor $76230$
Sign $1$
Analytic cond. $608.699$
Root an. cond. $24.6718$
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  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 2·13-s − 14-s + 16-s + 2·17-s + 4·19-s − 20-s + 8·23-s + 25-s + 2·26-s − 28-s + 6·29-s − 8·31-s + 32-s + 2·34-s + 35-s − 2·37-s + 4·38-s − 40-s + 2·41-s + 12·43-s + 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.312·41-s + 1.82·43-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76230\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(608.699\)
Root analytic conductor: \(24.6718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76230,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.671916839\)
\(L(\frac12)\) \(\approx\) \(4.671916839\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.99282708839015, −13.67417254408498, −12.90747620774937, −12.57995065866339, −12.27928063664935, −11.50163235465294, −11.16792443796565, −10.67213159187324, −10.21379138983774, −9.276912100097985, −9.183808014570162, −8.432155150856474, −7.677408024664838, −7.402649121478370, −6.798133395740113, −6.256637944304234, −5.618212077291013, −5.157341272020996, −4.598386131111294, −3.835276782402259, −3.439367285929143, −2.895231318894836, −2.227345859500888, −1.188355877426512, −0.7144086467350847, 0.7144086467350847, 1.188355877426512, 2.227345859500888, 2.895231318894836, 3.439367285929143, 3.835276782402259, 4.598386131111294, 5.157341272020996, 5.618212077291013, 6.256637944304234, 6.798133395740113, 7.402649121478370, 7.677408024664838, 8.432155150856474, 9.183808014570162, 9.276912100097985, 10.21379138983774, 10.67213159187324, 11.16792443796565, 11.50163235465294, 12.27928063664935, 12.57995065866339, 12.90747620774937, 13.67417254408498, 13.99282708839015

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