Properties

Label 2-420e2-1.1-c1-0-141
Degree $2$
Conductor $176400$
Sign $1$
Analytic cond. $1408.56$
Root an. cond. $37.5308$
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  + 6·11-s − 13-s − 3·17-s − 4·19-s + 3·23-s − 3·29-s + 5·31-s + 10·37-s + 9·41-s − 43-s + 9·53-s − 9·59-s − 11·61-s − 4·67-s − 12·71-s − 10·73-s + 10·79-s + 9·83-s − 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.80·11-s − 0.277·13-s − 0.727·17-s − 0.917·19-s + 0.625·23-s − 0.557·29-s + 0.898·31-s + 1.64·37-s + 1.40·41-s − 0.152·43-s + 1.23·53-s − 1.17·59-s − 1.40·61-s − 0.488·67-s − 1.42·71-s − 1.17·73-s + 1.12·79-s + 0.987·83-s − 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1408.56\)
Root analytic conductor: \(37.5308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{176400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.629718434\)
\(L(\frac12)\) \(\approx\) \(2.629718434\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.28984969561959, −12.67880151691307, −12.15696480316957, −11.82684458183704, −11.30382501502464, −10.86537632605946, −10.44893092862244, −9.732645183910576, −9.201807448659789, −9.076919784394219, −8.527716528379515, −7.796227249329106, −7.440104649365744, −6.748591419127302, −6.346030258377582, −6.071262792313385, −5.337094832876440, −4.502422716942608, −4.288842293095450, −3.874185436860548, −2.956158299950891, −2.577613818237959, −1.750227623774565, −1.235618919659125, −0.4887934393616325, 0.4887934393616325, 1.235618919659125, 1.750227623774565, 2.577613818237959, 2.956158299950891, 3.874185436860548, 4.288842293095450, 4.502422716942608, 5.337094832876440, 6.071262792313385, 6.346030258377582, 6.748591419127302, 7.440104649365744, 7.796227249329106, 8.527716528379515, 9.076919784394219, 9.201807448659789, 9.732645183910576, 10.44893092862244, 10.86537632605946, 11.30382501502464, 11.82684458183704, 12.15696480316957, 12.67880151691307, 13.28984969561959

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