Properties

Label 2-75712-1.1-c1-0-25
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  − 5-s − 7-s − 3·9-s − 2·11-s + 7·19-s + 3·23-s − 4·25-s + 9·29-s + 5·31-s + 35-s − 8·37-s + 10·41-s + 5·43-s + 3·45-s + 7·47-s + 49-s − 3·53-s + 2·55-s − 6·61-s + 3·63-s + 10·67-s + 4·71-s + 11·73-s + 2·77-s + 11·79-s + 9·81-s − 11·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s − 0.603·11-s + 1.60·19-s + 0.625·23-s − 4/5·25-s + 1.67·29-s + 0.898·31-s + 0.169·35-s − 1.31·37-s + 1.56·41-s + 0.762·43-s + 0.447·45-s + 1.02·47-s + 1/7·49-s − 0.412·53-s + 0.269·55-s − 0.768·61-s + 0.377·63-s + 1.22·67-s + 0.474·71-s + 1.28·73-s + 0.227·77-s + 1.23·79-s + 81-s − 1.20·83-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

\(L(1)\) \(\approx\) \(1.890323752\)
\(L(\frac12)\) \(\approx\) \(1.890323752\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−14.06875539628418, −13.73361399170799, −13.09502147535646, −12.29870854662715, −12.19335669096779, −11.62032215870070, −11.00409476046636, −10.69241860299644, −9.972046566946833, −9.531529915239061, −8.972831007382794, −8.447507881145682, −7.836466935318041, −7.574679369170327, −6.816774960824575, −6.296185077110478, −5.643992428052838, −5.223032168254610, −4.640937024555668, −3.873190516519837, −3.249557860025286, −2.791622549006066, −2.252001728592658, −1.044747364811460, −0.5330544712461626, 0.5330544712461626, 1.044747364811460, 2.252001728592658, 2.791622549006066, 3.249557860025286, 3.873190516519837, 4.640937024555668, 5.223032168254610, 5.643992428052838, 6.296185077110478, 6.816774960824575, 7.574679369170327, 7.836466935318041, 8.447507881145682, 8.972831007382794, 9.531529915239061, 9.972046566946833, 10.69241860299644, 11.00409476046636, 11.62032215870070, 12.19335669096779, 12.29870854662715, 13.09502147535646, 13.73361399170799, 14.06875539628418

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