Properties

Label 2-75712-1.1-c1-0-50
Degree $2$
Conductor $75712$
Sign $1$
Analytic cond. $604.563$
Root an. cond. $24.5878$
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-s + 4·5-s − 7-s − 2·9-s + 11-s + 4·15-s + 4·17-s − 2·19-s − 21-s + 7·23-s + 11·25-s − 5·27-s + 8·29-s + 3·31-s + 33-s − 4·35-s + 7·37-s + 7·41-s − 8·43-s − 8·45-s + 3·47-s + 49-s + 4·51-s + 4·55-s − 2·57-s + 6·59-s + 13·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 1.03·15-s + 0.970·17-s − 0.458·19-s − 0.218·21-s + 1.45·23-s + 11/5·25-s − 0.962·27-s + 1.48·29-s + 0.538·31-s + 0.174·33-s − 0.676·35-s + 1.15·37-s + 1.09·41-s − 1.21·43-s − 1.19·45-s + 0.437·47-s + 1/7·49-s + 0.560·51-s + 0.539·55-s − 0.264·57-s + 0.781·59-s + 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75712\)    =    \(2^{6} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(604.563\)
Root analytic conductor: \(24.5878\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.567572196\)
\(L(\frac12)\) \(\approx\) \(5.567572196\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−14.03706654038098, −13.53393352856822, −13.25426712311291, −12.73615197467893, −12.16257880815569, −11.57970470862775, −10.90305128168845, −10.42260627052081, −9.923595676697207, −9.474852963343540, −9.121232512445896, −8.534023669162447, −8.145055064509604, −7.317704785865966, −6.691020806800676, −6.233227820312667, −5.846451879596042, −5.166872159603785, −4.783703323686255, −3.801727315961331, −3.141992271332642, −2.572981398939446, −2.304704101025586, −1.278862539684480, −0.8096298412605660, 0.8096298412605660, 1.278862539684480, 2.304704101025586, 2.572981398939446, 3.141992271332642, 3.801727315961331, 4.783703323686255, 5.166872159603785, 5.846451879596042, 6.233227820312667, 6.691020806800676, 7.317704785865966, 8.145055064509604, 8.534023669162447, 9.121232512445896, 9.474852963343540, 9.923595676697207, 10.42260627052081, 10.90305128168845, 11.57970470862775, 12.16257880815569, 12.73615197467893, 13.25426712311291, 13.53393352856822, 14.03706654038098

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