Properties

Label 2-229320-1.1-c1-0-102
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 + 13-s + 2·17-s − 4·23-s + 25-s + 2·29-s + 8·31-s − 6·37-s + 2·41-s + 4·43-s + 8·47-s + 6·53-s + 8·59-s − 6·61-s − 65-s − 4·67-s + 2·73-s − 8·79-s − 12·83-s − 2·85-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.277·13-s + 0.485·17-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 0.824·53-s + 1.04·59-s − 0.768·61-s − 0.124·65-s − 0.488·67-s + 0.234·73-s − 0.900·79-s − 1.31·83-s − 0.216·85-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.17091000343854, −12.59844348214132, −12.15574438744333, −11.84889934937990, −11.43069939531997, −10.74521160548394, −10.40053788169649, −10.00723508352912, −9.418470113480854, −8.904784372767104, −8.333857905860192, −8.124238777003716, −7.465361474334382, −7.017008650705207, −6.564787220884943, −5.786059032836837, −5.672628726157831, −4.865259775547520, −4.279769404078341, −3.984182192452705, −3.283515151564565, −2.752530388523238, −2.192843975878171, −1.360738397166694, −0.8294308950780113, 0, 0.8294308950780113, 1.360738397166694, 2.192843975878171, 2.752530388523238, 3.283515151564565, 3.984182192452705, 4.279769404078341, 4.865259775547520, 5.672628726157831, 5.786059032836837, 6.564787220884943, 7.017008650705207, 7.465361474334382, 8.124238777003716, 8.333857905860192, 8.904784372767104, 9.418470113480854, 10.00723508352912, 10.40053788169649, 10.74521160548394, 11.43069939531997, 11.84889934937990, 12.15574438744333, 12.59844348214132, 13.17091000343854

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