Properties

Label 2-229320-1.1-c1-0-11
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 − 4·11-s + 13-s + 6·17-s + 4·19-s − 4·23-s + 25-s + 10·29-s − 8·31-s + 6·37-s − 2·41-s − 4·43-s − 2·53-s + 4·55-s − 4·59-s − 10·61-s − 65-s + 4·67-s − 12·71-s − 10·73-s − 16·83-s − 6·85-s + 6·89-s − 4·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.274·53-s + 0.539·55-s − 0.520·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 1.75·83-s − 0.650·85-s + 0.635·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(1.325951878\)
\(L(\frac12)\) \(\approx\) \(1.325951878\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
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 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−12.91495353526281, −12.40161667826045, −11.90702521675606, −11.79561945039829, −10.91327732653950, −10.69954979633622, −10.10267584672188, −9.773975040475718, −9.276803609663443, −8.486775976918194, −8.230777763659413, −7.698036724682120, −7.407950048902471, −6.850388003579326, −6.055490555479480, −5.748495423312368, −5.215335158224427, −4.701717789686729, −4.155174446243092, −3.471257519819877, −2.984178785041681, −2.665223438329108, −1.655253284128733, −1.189593058031004, −0.3345940157640187, 0.3345940157640187, 1.189593058031004, 1.655253284128733, 2.665223438329108, 2.984178785041681, 3.471257519819877, 4.155174446243092, 4.701717789686729, 5.215335158224427, 5.748495423312368, 6.055490555479480, 6.850388003579326, 7.407950048902471, 7.698036724682120, 8.230777763659413, 8.486775976918194, 9.276803609663443, 9.773975040475718, 10.10267584672188, 10.69954979633622, 10.91327732653950, 11.79561945039829, 11.90702521675606, 12.40161667826045, 12.91495353526281

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