Properties

Label 2-229320-1.1-c1-0-105
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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  − 5-s − 11-s + 13-s + 3·17-s + 8·19-s − 5·23-s + 25-s − 6·29-s + 2·31-s + 7·37-s + 3·41-s − 2·43-s − 6·47-s − 6·53-s + 55-s − 5·59-s + 14·61-s − 65-s − 67-s + 9·73-s + 3·79-s + 6·83-s − 3·85-s − 3·89-s − 8·95-s − 97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 0.277·13-s + 0.727·17-s + 1.83·19-s − 1.04·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 1.15·37-s + 0.468·41-s − 0.304·43-s − 0.875·47-s − 0.824·53-s + 0.134·55-s − 0.650·59-s + 1.79·61-s − 0.124·65-s − 0.122·67-s + 1.05·73-s + 0.337·79-s + 0.658·83-s − 0.325·85-s − 0.317·89-s − 0.820·95-s − 0.101·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.25433218807666, −12.60790814540261, −12.25353683085150, −11.66502841710763, −11.42102511792842, −10.96868261488257, −10.30797541462002, −9.846466237518313, −9.486090065177263, −9.084129374514406, −8.222918265008250, −7.938578232471427, −7.652444459633200, −7.081593547911363, −6.443846986291498, −5.979634763240377, −5.311736664480447, −5.141831111224776, −4.311249790427370, −3.801340484247574, −3.347144024035568, −2.817908308256042, −2.133731383986067, −1.370727798493825, −0.8350805036272671, 0, 0.8350805036272671, 1.370727798493825, 2.133731383986067, 2.817908308256042, 3.347144024035568, 3.801340484247574, 4.311249790427370, 5.141831111224776, 5.311736664480447, 5.979634763240377, 6.443846986291498, 7.081593547911363, 7.652444459633200, 7.938578232471427, 8.222918265008250, 9.084129374514406, 9.486090065177263, 9.846466237518313, 10.30797541462002, 10.96868261488257, 11.42102511792842, 11.66502841710763, 12.25353683085150, 12.60790814540261, 13.25433218807666

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