Properties

Label 2-75712-1.1-c1-0-46
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  − 2·5-s − 7-s − 3·9-s + 4·11-s + 2·17-s − 4·19-s + 4·23-s − 25-s + 2·29-s + 2·35-s + 2·37-s + 2·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s + 2·53-s − 8·55-s + 4·59-s − 14·61-s + 3·63-s − 12·67-s + 8·71-s − 14·73-s − 4·77-s + 12·79-s + 9·81-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-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 + 0.274·53-s − 1.07·55-s + 0.520·59-s − 1.79·61-s + 0.377·63-s − 1.46·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s + 1.35·79-s + 81-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.37796198882090, −13.89777676749566, −13.31581218593354, −12.67973182580703, −12.25304176453250, −11.80884271234814, −11.31771593813609, −11.01220159573242, −10.31775550237983, −9.734054823961064, −9.094566451636711, −8.767282688248230, −8.269640955601634, −7.632943632280902, −7.193438672296683, −6.452264372390881, −6.127240380940390, −5.545514675445481, −4.653581511956927, −4.314299689154804, −3.543981909273187, −3.202720660430773, −2.495902709976859, −1.607692229794035, −0.7786653953247407, 0, 0.7786653953247407, 1.607692229794035, 2.495902709976859, 3.202720660430773, 3.543981909273187, 4.314299689154804, 4.653581511956927, 5.545514675445481, 6.127240380940390, 6.452264372390881, 7.193438672296683, 7.632943632280902, 8.269640955601634, 8.767282688248230, 9.094566451636711, 9.734054823961064, 10.31775550237983, 11.01220159573242, 11.31771593813609, 11.80884271234814, 12.25304176453250, 12.67973182580703, 13.31581218593354, 13.89777676749566, 14.37796198882090

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