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Properties

Label 2-420e2-1.1-c1-0-23
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  − 5·13-s − 19-s − 7·31-s − 10·37-s − 5·43-s + 13·61-s − 5·67-s + 10·73-s + 4·79-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.38·13-s − 0.229·19-s − 1.25·31-s − 1.64·37-s − 0.762·43-s + 1.66·61-s − 0.610·67-s + 1.17·73-s + 0.450·79-s − 0.507·97-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6850779414\)
\(L(\frac12)\) \(\approx\) \(0.6850779414\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.04415090530453, −12.73825990133772, −12.25413346063013, −11.78333057826353, −11.40513736983103, −10.71752821704596, −10.34227962880383, −9.918578313582761, −9.336514058801859, −8.970026387219508, −8.402101500808210, −7.816200818325224, −7.432060092338279, −6.760518897404306, −6.633934437896016, −5.691453282237549, −5.241387398375875, −4.942782582925688, −4.195806320516564, −3.659021587430011, −3.125254005884935, −2.348180037537912, −1.996468301314444, −1.216978692273296, −0.2372914341912898, 0.2372914341912898, 1.216978692273296, 1.996468301314444, 2.348180037537912, 3.125254005884935, 3.659021587430011, 4.195806320516564, 4.942782582925688, 5.241387398375875, 5.691453282237549, 6.633934437896016, 6.760518897404306, 7.432060092338279, 7.816200818325224, 8.402101500808210, 8.970026387219508, 9.336514058801859, 9.918578313582761, 10.34227962880383, 10.71752821704596, 11.40513736983103, 11.78333057826353, 12.25413346063013, 12.73825990133772, 13.04415090530453

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