Properties

Label 2-75712-1.1-c1-0-19
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 7-s − 3·9-s − 2·11-s − 4·17-s − 19-s − 7·23-s + 4·25-s − 7·29-s + 5·31-s + 3·35-s + 4·37-s + 6·41-s − 9·43-s + 9·45-s + 7·47-s + 49-s − 11·53-s + 6·55-s + 2·61-s + 3·63-s − 10·67-s − 7·73-s + 2·77-s + 79-s + 9·81-s − 11·83-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s − 9-s − 0.603·11-s − 0.970·17-s − 0.229·19-s − 1.45·23-s + 4/5·25-s − 1.29·29-s + 0.898·31-s + 0.507·35-s + 0.657·37-s + 0.937·41-s − 1.37·43-s + 1.34·45-s + 1.02·47-s + 1/7·49-s − 1.51·53-s + 0.809·55-s + 0.256·61-s + 0.377·63-s − 1.22·67-s − 0.819·73-s + 0.227·77-s + 0.112·79-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: \(1\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.45461864936870, −13.72646540567056, −13.30147254347342, −12.82259337952359, −12.14334881971562, −11.83941378466625, −11.33511968411311, −10.92104443804169, −10.43987458937601, −9.705451817101760, −9.237588445567016, −8.523641733183708, −8.227732961992044, −7.726453423503517, −7.258549217785760, −6.549771286682929, −5.961448850577966, −5.583474292573412, −4.606073932685489, −4.337001290507574, −3.647076586748627, −3.083086413199201, −2.494980399427662, −1.780482681136548, −0.5061845841716397, 0, 0.5061845841716397, 1.780482681136548, 2.494980399427662, 3.083086413199201, 3.647076586748627, 4.337001290507574, 4.606073932685489, 5.583474292573412, 5.961448850577966, 6.549771286682929, 7.258549217785760, 7.726453423503517, 8.227732961992044, 8.523641733183708, 9.237588445567016, 9.705451817101760, 10.43987458937601, 10.92104443804169, 11.33511968411311, 11.83941378466625, 12.14334881971562, 12.82259337952359, 13.30147254347342, 13.72646540567056, 14.45461864936870

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