Properties

Label 2-229320-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 11-s − 13-s − 3·17-s − 6·19-s + 7·23-s + 25-s − 6·29-s + 2·31-s + 37-s + 7·41-s + 8·43-s + 2·47-s − 13·53-s − 55-s − 8·59-s + 7·61-s − 65-s + 12·67-s − 71-s + 10·73-s + 3·79-s + 8·83-s − 3·85-s + 13·89-s − 6·95-s − 7·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.45·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.164·37-s + 1.09·41-s + 1.21·43-s + 0.291·47-s − 1.78·53-s − 0.134·55-s − 1.04·59-s + 0.896·61-s − 0.124·65-s + 1.46·67-s − 0.118·71-s + 1.17·73-s + 0.337·79-s + 0.878·83-s − 0.325·85-s + 1.37·89-s − 0.615·95-s − 0.710·97-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: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.933392295\)
\(L(\frac12)\) \(\approx\) \(1.933392295\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.86281282758829, −12.60252349419021, −12.19683750822636, −11.24767506987942, −11.05117803014032, −10.78955938314883, −10.17381142187572, −9.541937013448928, −9.155305937166999, −8.922719545876930, −8.141238409768068, −7.786936676117486, −7.237738773494010, −6.555221845104404, −6.393711525050117, −5.739505581474812, −5.078823609953377, −4.825022445909721, −4.085740874889330, −3.702075912206393, −2.778653063402571, −2.474120207450105, −1.911461135352651, −1.148405688101009, −0.3967012058199847, 0.3967012058199847, 1.148405688101009, 1.911461135352651, 2.474120207450105, 2.778653063402571, 3.702075912206393, 4.085740874889330, 4.825022445909721, 5.078823609953377, 5.739505581474812, 6.393711525050117, 6.555221845104404, 7.237738773494010, 7.786936676117486, 8.141238409768068, 8.922719545876930, 9.155305937166999, 9.541937013448928, 10.17381142187572, 10.78955938314883, 11.05117803014032, 11.24767506987942, 12.19683750822636, 12.60252349419021, 12.86281282758829

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