Properties

Label 2-229320-1.1-c1-0-37
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 − 2·17-s − 4·23-s + 25-s + 10·29-s + 4·31-s + 6·37-s − 2·41-s − 4·43-s + 14·53-s + 4·55-s + 4·59-s + 10·61-s − 65-s − 4·67-s + 2·73-s + 12·83-s + 2·85-s − 2·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.92·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s − 0.124·65-s − 0.488·67-s + 0.234·73-s + 1.31·83-s + 0.216·85-s − 0.211·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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

\(L(1)\) \(\approx\) \(2.193439473\)
\(L(\frac12)\) \(\approx\) \(2.193439473\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−12.97670312999805, −12.48944228553285, −11.79355819512789, −11.75571463085393, −11.09591311858076, −10.45649380166687, −10.24550907584072, −9.857997030082414, −9.081380618343471, −8.602321997722958, −8.112611890194291, −7.964164826806378, −7.208619479572860, −6.770596199767648, −6.258171305008221, −5.716751591930082, −5.148990179244318, −4.631668447309145, −4.218932338460882, −3.574308018279737, −2.956258042465454, −2.450005885031661, −1.964387898393438, −0.9199351399347756, −0.5011539515759570, 0.5011539515759570, 0.9199351399347756, 1.964387898393438, 2.450005885031661, 2.956258042465454, 3.574308018279737, 4.218932338460882, 4.631668447309145, 5.148990179244318, 5.716751591930082, 6.258171305008221, 6.770596199767648, 7.208619479572860, 7.964164826806378, 8.112611890194291, 8.602321997722958, 9.081380618343471, 9.857997030082414, 10.24550907584072, 10.45649380166687, 11.09591311858076, 11.75571463085393, 11.79355819512789, 12.48944228553285, 12.97670312999805

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