Properties

Label 2-15210-1.1-c1-0-18
Degree $2$
Conductor $15210$
Sign $1$
Analytic cond. $121.452$
Root an. cond. $11.0205$
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 + 4·7-s + 8-s − 10-s − 5·11-s + 4·14-s + 16-s − 2·17-s + 6·19-s − 20-s − 5·22-s + 9·23-s + 25-s + 4·28-s − 5·29-s + 3·31-s + 32-s − 2·34-s − 4·35-s + 5·37-s + 6·38-s − 40-s − 2·41-s − 7·43-s − 5·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.223·20-s − 1.06·22-s + 1.87·23-s + 1/5·25-s + 0.755·28-s − 0.928·29-s + 0.538·31-s + 0.176·32-s − 0.342·34-s − 0.676·35-s + 0.821·37-s + 0.973·38-s − 0.158·40-s − 0.312·41-s − 1.06·43-s − 0.753·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15210\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(121.452\)
Root analytic conductor: \(11.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15210} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15210,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.891414669\)
\(L(\frac12)\) \(\approx\) \(3.891414669\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−15.83264834850650, −15.34270881562684, −14.93398295857367, −14.51982868172206, −13.58375436832218, −13.44810811812387, −12.78295903043591, −12.02945800881518, −11.52428841388719, −11.07170477027213, −10.67016917089619, −9.936173915420598, −9.024805820719334, −8.438911270942159, −7.701309663024329, −7.480657117628339, −6.797668901051893, −5.673138685170485, −5.234048491061690, −4.806871576557714, −4.162843578517969, −3.158801942787603, −2.642838381950683, −1.718535751580558, −0.7871941586241400, 0.7871941586241400, 1.718535751580558, 2.642838381950683, 3.158801942787603, 4.162843578517969, 4.806871576557714, 5.234048491061690, 5.673138685170485, 6.797668901051893, 7.480657117628339, 7.701309663024329, 8.438911270942159, 9.024805820719334, 9.936173915420598, 10.67016917089619, 11.07170477027213, 11.52428841388719, 12.02945800881518, 12.78295903043591, 13.44810811812387, 13.58375436832218, 14.51982868172206, 14.93398295857367, 15.34270881562684, 15.83264834850650

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