Properties

Label 2-54418-1.1-c1-0-2
Degree $2$
Conductor $54418$
Sign $1$
Analytic cond. $434.529$
Root an. cond. $20.8453$
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  + 2-s + 4-s + 3·5-s − 7-s + 8-s − 3·9-s + 3·10-s − 14-s + 16-s + 3·17-s − 3·18-s + 8·19-s + 3·20-s − 23-s + 4·25-s − 28-s − 5·29-s + 8·31-s + 32-s + 3·34-s − 3·35-s − 3·36-s − 5·37-s + 8·38-s + 3·40-s − 5·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.948·10-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.83·19-s + 0.670·20-s − 0.208·23-s + 4/5·25-s − 0.188·28-s − 0.928·29-s + 1.43·31-s + 0.176·32-s + 0.514·34-s − 0.507·35-s − 1/2·36-s − 0.821·37-s + 1.29·38-s + 0.474·40-s − 0.780·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54418\)    =    \(2 \cdot 7 \cdot 13^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(434.529\)
Root analytic conductor: \(20.8453\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{54418} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 54418,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.181996405\)
\(L(\frac12)\) \(\approx\) \(5.181996405\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.20441060810315, −13.90672377071760, −13.40063752005193, −13.27185177618893, −12.27776366321084, −11.94231710649981, −11.59941163729232, −10.80750179639052, −10.26574654898732, −9.819271559972655, −9.386364765212082, −8.809344119755723, −8.063671461251195, −7.584618060408327, −6.735305648559674, −6.440814122286023, −5.634583585184042, −5.437470505782328, −5.053730705596414, −4.017067009266276, −3.277796540902445, −2.954742976222392, −2.218736429343807, −1.538443416852809, −0.6947341640455030, 0.6947341640455030, 1.538443416852809, 2.218736429343807, 2.954742976222392, 3.277796540902445, 4.017067009266276, 5.053730705596414, 5.437470505782328, 5.634583585184042, 6.440814122286023, 6.735305648559674, 7.584618060408327, 8.063671461251195, 8.809344119755723, 9.386364765212082, 9.819271559972655, 10.26574654898732, 10.80750179639052, 11.59941163729232, 11.94231710649981, 12.27776366321084, 13.27185177618893, 13.40063752005193, 13.90672377071760, 14.20441060810315

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