Properties

Label 2-53312-1.1-c1-0-0
Degree $2$
Conductor $53312$
Sign $1$
Analytic cond. $425.698$
Root an. cond. $20.6324$
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  − 3·3-s − 2·5-s + 6·9-s − 5·11-s − 3·13-s + 6·15-s + 17-s + 2·19-s + 8·23-s − 25-s − 9·27-s + 6·29-s − 4·31-s + 15·33-s − 8·37-s + 9·39-s − 6·41-s + 4·43-s − 12·45-s − 10·47-s − 3·51-s − 9·53-s + 10·55-s − 6·57-s − 4·59-s + 4·61-s + 6·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s + 2·9-s − 1.50·11-s − 0.832·13-s + 1.54·15-s + 0.242·17-s + 0.458·19-s + 1.66·23-s − 1/5·25-s − 1.73·27-s + 1.11·29-s − 0.718·31-s + 2.61·33-s − 1.31·37-s + 1.44·39-s − 0.937·41-s + 0.609·43-s − 1.78·45-s − 1.45·47-s − 0.420·51-s − 1.23·53-s + 1.34·55-s − 0.794·57-s − 0.520·59-s + 0.512·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53312\)    =    \(2^{6} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(425.698\)
Root analytic conductor: \(20.6324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 53312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1856117439\)
\(L(\frac12)\) \(\approx\) \(0.1856117439\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.55686181868722, −13.83903568295580, −13.20326474019582, −12.60492819478058, −12.47132239785690, −11.80230581168329, −11.41058249867521, −10.95005950263045, −10.42892066405044, −10.09812795616875, −9.440272904842737, −8.650899170621421, −7.939499811575638, −7.504864671446105, −7.035393624956508, −6.542370271635661, −5.765041501043960, −5.170447834983602, −4.922159030243845, −4.477403409044873, −3.425884252131551, −2.994840608736967, −1.949318495027014, −1.038163023892590, −0.1978726779821674, 0.1978726779821674, 1.038163023892590, 1.949318495027014, 2.994840608736967, 3.425884252131551, 4.477403409044873, 4.922159030243845, 5.170447834983602, 5.765041501043960, 6.542370271635661, 7.035393624956508, 7.504864671446105, 7.939499811575638, 8.650899170621421, 9.440272904842737, 10.09812795616875, 10.42892066405044, 10.95005950263045, 11.41058249867521, 11.80230581168329, 12.47132239785690, 12.60492819478058, 13.20326474019582, 13.83903568295580, 14.55686181868722

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