Properties

Label 2-75712-1.1-c1-0-9
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  + 2·5-s − 7-s − 3·9-s + 4·11-s − 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s − 2·35-s − 2·37-s − 2·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s − 6·53-s + 8·55-s + 6·61-s + 3·63-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s − 16·79-s + 9·81-s − 8·83-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.377·63-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s + 81-s − 0.878·83-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: $\chi_{75712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9006482330\)
\(L(\frac12)\) \(\approx\) \(0.9006482330\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.06929234842114, −13.41792820351619, −13.22831441724580, −12.69854964168413, −11.99008811816699, −11.42331636105186, −11.24627035055101, −10.45461573590792, −10.04731260141984, −9.441928800920673, −8.974932202631965, −8.566843638996432, −8.194398793541814, −7.198447471395431, −6.599068927455776, −6.300438417881343, −5.957491663514909, −5.210301008323480, −4.470966572206251, −4.073841272954043, −3.285551625424022, −2.619801442249240, −1.991468591533222, −1.538260187510115, −0.2894098599452606, 0.2894098599452606, 1.538260187510115, 1.991468591533222, 2.619801442249240, 3.285551625424022, 4.073841272954043, 4.470966572206251, 5.210301008323480, 5.957491663514909, 6.300438417881343, 6.599068927455776, 7.198447471395431, 8.194398793541814, 8.566843638996432, 8.974932202631965, 9.441928800920673, 10.04731260141984, 10.45461573590792, 11.24627035055101, 11.42331636105186, 11.99008811816699, 12.69854964168413, 13.22831441724580, 13.41792820351619, 14.06929234842114

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