Properties

Label 2-75712-1.1-c1-0-40
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

Downloads

Learn more

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 \( 1 - 13 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.46770121291746, −13.95661137712450, −13.09921409012825, −12.88891584651907, −12.24094492925722, −11.62469146872765, −11.23800427466153, −11.10382235324461, −10.60611472830053, −9.436921449153117, −9.236272618981343, −8.774858360617892, −8.094254841915279, −7.719638645435805, −7.242385761704525, −6.608817539493911, −6.065231742572852, −5.300419060630825, −4.885485695728580, −4.096991808488740, −3.805368346765951, −3.070531079433432, −2.506153544352445, −1.625892277778001, −0.7408344975142312, 0, 0.7408344975142312, 1.625892277778001, 2.506153544352445, 3.070531079433432, 3.805368346765951, 4.096991808488740, 4.885485695728580, 5.300419060630825, 6.065231742572852, 6.608817539493911, 7.242385761704525, 7.719638645435805, 8.094254841915279, 8.774858360617892, 9.236272618981343, 9.436921449153117, 10.60611472830053, 11.10382235324461, 11.23800427466153, 11.62469146872765, 12.24094492925722, 12.88891584651907, 13.09921409012825, 13.95661137712450, 14.46770121291746

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