Properties

Label 2-229320-1.1-c1-0-58
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 − 3·11-s + 13-s − 5·17-s − 8·19-s + 5·23-s + 25-s − 10·29-s − 6·31-s + 3·37-s + 7·41-s − 2·43-s − 6·47-s + 10·53-s + 3·55-s + 59-s − 2·61-s − 65-s + 5·67-s + 16·71-s − 11·73-s − 11·79-s + 14·83-s + 5·85-s + 89-s + 8·95-s + 19·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s + 0.277·13-s − 1.21·17-s − 1.83·19-s + 1.04·23-s + 1/5·25-s − 1.85·29-s − 1.07·31-s + 0.493·37-s + 1.09·41-s − 0.304·43-s − 0.875·47-s + 1.37·53-s + 0.404·55-s + 0.130·59-s − 0.256·61-s − 0.124·65-s + 0.610·67-s + 1.89·71-s − 1.28·73-s − 1.23·79-s + 1.53·83-s + 0.542·85-s + 0.105·89-s + 0.820·95-s + 1.92·97-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: \(1\)
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 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−13.10207226498860, −12.94827136155543, −12.37587340380199, −11.60881418605943, −11.19539733639783, −10.88844965513498, −10.60210835618282, −9.945887206405143, −9.249867397081419, −8.965784830121965, −8.488086310801670, −8.020471150561295, −7.381815000823637, −7.151120485880614, −6.403770849616178, −6.110440917524948, −5.325385409270201, −5.008877788744438, −4.294439246967240, −3.929972800443429, −3.399782308949478, −2.543457644129919, −2.237979349990821, −1.594086516933097, −0.5762506718582921, 0, 0.5762506718582921, 1.594086516933097, 2.237979349990821, 2.543457644129919, 3.399782308949478, 3.929972800443429, 4.294439246967240, 5.008877788744438, 5.325385409270201, 6.110440917524948, 6.403770849616178, 7.151120485880614, 7.381815000823637, 8.020471150561295, 8.488086310801670, 8.965784830121965, 9.249867397081419, 9.945887206405143, 10.60210835618282, 10.88844965513498, 11.19539733639783, 11.60881418605943, 12.37587340380199, 12.94827136155543, 13.10207226498860

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