Properties

Label 2-229320-1.1-c1-0-72
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 − 13-s + 5·17-s + 5·19-s + 8·23-s + 25-s + 2·29-s − 6·31-s + 6·37-s + 11·41-s − 11·43-s − 2·53-s + 2·59-s + 14·61-s − 65-s + 2·67-s + 13·71-s − 7·73-s + 7·79-s − 11·83-s + 5·85-s − 89-s + 5·95-s + 17·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.277·13-s + 1.21·17-s + 1.14·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s − 1.07·31-s + 0.986·37-s + 1.71·41-s − 1.67·43-s − 0.274·53-s + 0.260·59-s + 1.79·61-s − 0.124·65-s + 0.244·67-s + 1.54·71-s − 0.819·73-s + 0.787·79-s − 1.20·83-s + 0.542·85-s − 0.105·89-s + 0.512·95-s + 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(4.224659833\)
\(L(\frac12)\) \(\approx\) \(4.224659833\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.01181031869541, −12.62014704782947, −11.96060499840191, −11.42601520491444, −11.27022548333598, −10.47304245234413, −10.18108083124432, −9.492978756846634, −9.407396427622784, −8.805107461702276, −8.115901322508838, −7.771646306946941, −7.142927823360324, −6.892571214814037, −6.160738226283524, −5.651419128933144, −5.160213772317760, −4.920435245121451, −4.079388932985349, −3.473395083018733, −3.022175807239141, −2.502046776743133, −1.760703258141923, −1.048104391069535, −0.6659920123231517, 0.6659920123231517, 1.048104391069535, 1.760703258141923, 2.502046776743133, 3.022175807239141, 3.473395083018733, 4.079388932985349, 4.920435245121451, 5.160213772317760, 5.651419128933144, 6.160738226283524, 6.892571214814037, 7.142927823360324, 7.771646306946941, 8.115901322508838, 8.805107461702276, 9.407396427622784, 9.492978756846634, 10.18108083124432, 10.47304245234413, 11.27022548333598, 11.42601520491444, 11.96060499840191, 12.62014704782947, 13.01181031869541

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