Properties

Label 2-229320-1.1-c1-0-12
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 + 11-s − 13-s − 7·17-s + 2·19-s − 3·23-s + 25-s − 6·29-s + 8·31-s − 37-s − 9·41-s + 2·43-s − 8·47-s + 6·53-s + 55-s − 3·59-s + 14·61-s − 65-s + 15·67-s + 8·71-s − 73-s − 79-s − 8·83-s − 7·85-s − 15·89-s + 2·95-s − 3·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s − 0.277·13-s − 1.69·17-s + 0.458·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.164·37-s − 1.40·41-s + 0.304·43-s − 1.16·47-s + 0.824·53-s + 0.134·55-s − 0.390·59-s + 1.79·61-s − 0.124·65-s + 1.83·67-s + 0.949·71-s − 0.117·73-s − 0.112·79-s − 0.878·83-s − 0.759·85-s − 1.58·89-s + 0.205·95-s − 0.304·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.553628555\)
\(L(\frac12)\) \(\approx\) \(1.553628555\)
\(L(\frac{3}{2})\) 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 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−13.08277581841704, −12.44348927139096, −12.02015193503476, −11.42058727210735, −11.21174188517557, −10.64013682223930, −9.913087493263745, −9.800548748969185, −9.276645773658152, −8.618130848952298, −8.332380707011875, −7.834422981896274, −6.962148235698441, −6.788090639230358, −6.398019905092432, −5.577171095747304, −5.317605890962840, −4.657813018973675, −4.121721218133559, −3.683207890129475, −2.901596264331054, −2.356285426521097, −1.887267606608970, −1.221580132761691, −0.3424926831052709, 0.3424926831052709, 1.221580132761691, 1.887267606608970, 2.356285426521097, 2.901596264331054, 3.683207890129475, 4.121721218133559, 4.657813018973675, 5.317605890962840, 5.577171095747304, 6.398019905092432, 6.788090639230358, 6.962148235698441, 7.834422981896274, 8.332380707011875, 8.618130848952298, 9.276645773658152, 9.800548748969185, 9.913087493263745, 10.64013682223930, 11.21174188517557, 11.42058727210735, 12.02015193503476, 12.44348927139096, 13.08277581841704

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