Properties

Label 2-229320-1.1-c1-0-114
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 − 11-s + 13-s + 2·17-s + 2·19-s + 25-s − 4·29-s + 4·31-s + 7·37-s − 10·41-s + 4·43-s + 12·47-s − 5·53-s − 55-s − 15·59-s + 3·61-s + 65-s + 3·67-s + 9·71-s + 9·73-s + 79-s − 14·83-s + 2·85-s − 13·89-s + 2·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.301·11-s + 0.277·13-s + 0.485·17-s + 0.458·19-s + 1/5·25-s − 0.742·29-s + 0.718·31-s + 1.15·37-s − 1.56·41-s + 0.609·43-s + 1.75·47-s − 0.686·53-s − 0.134·55-s − 1.95·59-s + 0.384·61-s + 0.124·65-s + 0.366·67-s + 1.06·71-s + 1.05·73-s + 0.112·79-s − 1.53·83-s + 0.216·85-s − 1.37·89-s + 0.205·95-s + 0.203·97-s + 0.0995·101-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 \( 1 + T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
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

−13.03421100621596, −12.84212871930277, −12.22262726821430, −11.81470040936607, −11.27885442881815, −10.81424766976025, −10.38839119361073, −9.890387332710838, −9.364647869330299, −9.155847854699148, −8.333364355399596, −8.056129392334885, −7.491554817863182, −7.013456563558558, −6.410010315531129, −5.971154848301923, −5.454460026958441, −5.058829517141997, −4.385224925834800, −3.874209729464352, −3.227729896136495, −2.724785999314360, −2.155296440585644, −1.417404061118479, −0.9099831251640267, 0, 0.9099831251640267, 1.417404061118479, 2.155296440585644, 2.724785999314360, 3.227729896136495, 3.874209729464352, 4.385224925834800, 5.058829517141997, 5.454460026958441, 5.971154848301923, 6.410010315531129, 7.013456563558558, 7.491554817863182, 8.056129392334885, 8.333364355399596, 9.155847854699148, 9.364647869330299, 9.890387332710838, 10.38839119361073, 10.81424766976025, 11.27885442881815, 11.81470040936607, 12.22262726821430, 12.84212871930277, 13.03421100621596

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