Properties

Label 2-229320-1.1-c1-0-7
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 − 3·11-s + 13-s − 2·17-s + 4·19-s − 4·23-s + 25-s − 4·29-s + 3·37-s − 2·41-s − 2·43-s − 6·47-s + 13·53-s + 3·55-s − 5·59-s + 7·61-s − 65-s − 13·67-s − 5·71-s − 11·73-s + 13·79-s − 4·83-s + 2·85-s + 89-s − 4·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.904·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.742·29-s + 0.493·37-s − 0.312·41-s − 0.304·43-s − 0.875·47-s + 1.78·53-s + 0.404·55-s − 0.650·59-s + 0.896·61-s − 0.124·65-s − 1.58·67-s − 0.593·71-s − 1.28·73-s + 1.46·79-s − 0.439·83-s + 0.216·85-s + 0.105·89-s − 0.410·95-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: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9630284218\)
\(L(\frac12)\) \(\approx\) \(0.9630284218\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−13.06985863158104, −12.41910926297992, −11.94069431485908, −11.56331933527284, −11.15346011507898, −10.59523925258455, −10.12165015197354, −9.812443214410307, −9.062146300436915, −8.742175517101690, −8.185111925573527, −7.553973661684834, −7.519393933371850, −6.752332825302451, −6.224470545227515, −5.646757682688950, −5.258592849869783, −4.636235927658695, −4.148077307277094, −3.550161599686657, −3.062044662503448, −2.440657525918556, −1.844262909455408, −1.115051773845010, −0.2869033941993053, 0.2869033941993053, 1.115051773845010, 1.844262909455408, 2.440657525918556, 3.062044662503448, 3.550161599686657, 4.148077307277094, 4.636235927658695, 5.258592849869783, 5.646757682688950, 6.224470545227515, 6.752332825302451, 7.519393933371850, 7.553973661684834, 8.185111925573527, 8.742175517101690, 9.062146300436915, 9.812443214410307, 10.12165015197354, 10.59523925258455, 11.15346011507898, 11.56331933527284, 11.94069431485908, 12.41910926297992, 13.06985863158104

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