Properties

Label 2-229320-1.1-c1-0-98
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 − 2·11-s − 13-s − 2·17-s + 6·19-s + 4·23-s + 25-s + 8·29-s − 8·31-s + 2·37-s + 6·41-s − 4·43-s + 4·47-s − 6·53-s + 2·55-s + 12·61-s + 65-s − 6·67-s − 12·71-s + 6·73-s + 6·83-s + 2·85-s + 6·89-s − 6·95-s − 14·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.603·11-s − 0.277·13-s − 0.485·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 1.48·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.583·47-s − 0.824·53-s + 0.269·55-s + 1.53·61-s + 0.124·65-s − 0.733·67-s − 1.42·71-s + 0.702·73-s + 0.658·83-s + 0.216·85-s + 0.635·89-s − 0.615·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.11527748138325, −12.72014415815013, −12.25193613156827, −11.69847311751142, −11.39066656425211, −10.83960867695952, −10.43026942988554, −9.955825723164384, −9.275510549222184, −9.090493549530351, −8.420206900894132, −7.895997875390625, −7.541977491035282, −6.997960371515373, −6.658208817148745, −5.853654525844027, −5.460218378682238, −4.847128563712251, −4.560765947744404, −3.782826492700428, −3.281429656532714, −2.745862614465912, −2.252873066456787, −1.358128546563839, −0.7912433043592371, 0, 0.7912433043592371, 1.358128546563839, 2.252873066456787, 2.745862614465912, 3.281429656532714, 3.782826492700428, 4.560765947744404, 4.847128563712251, 5.460218378682238, 5.853654525844027, 6.658208817148745, 6.997960371515373, 7.541977491035282, 7.895997875390625, 8.420206900894132, 9.090493549530351, 9.275510549222184, 9.955825723164384, 10.43026942988554, 10.83960867695952, 11.39066656425211, 11.69847311751142, 12.25193613156827, 12.72014415815013, 13.11527748138325

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