Properties

Label 2-30960-1.1-c1-0-3
Degree $2$
Conductor $30960$
Sign $1$
Analytic cond. $247.216$
Root an. cond. $15.7231$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·13-s + 2·17-s − 8·23-s + 25-s + 2·29-s + 8·31-s − 6·37-s − 2·41-s − 43-s − 7·49-s − 2·53-s − 8·59-s − 10·61-s + 2·65-s + 4·67-s + 14·73-s − 8·79-s + 4·83-s − 2·85-s + 6·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.554·13-s + 0.485·17-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.312·41-s − 0.152·43-s − 49-s − 0.274·53-s − 1.04·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 1.63·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(247.216\)
Root analytic conductor: \(15.7231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{30960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.276832430\)
\(L(\frac12)\) \(\approx\) \(1.276832430\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−15.26169834254907, −14.45614278696291, −14.04433682819312, −13.71416630774887, −12.83811092196647, −12.39738185863570, −11.86557637584724, −11.60040236155155, −10.73938634762257, −10.21274923566438, −9.848265062180391, −9.152480126050646, −8.497425322558534, −7.825484481976224, −7.715980510485031, −6.694945639720938, −6.360340051371965, −5.599302615026062, −4.867444451035481, −4.427202084400599, −3.619445890227339, −3.074362252607159, −2.242753643227377, −1.484167429790019, −0.4313100689766457, 0.4313100689766457, 1.484167429790019, 2.242753643227377, 3.074362252607159, 3.619445890227339, 4.427202084400599, 4.867444451035481, 5.599302615026062, 6.360340051371965, 6.694945639720938, 7.715980510485031, 7.825484481976224, 8.497425322558534, 9.152480126050646, 9.848265062180391, 10.21274923566438, 10.73938634762257, 11.60040236155155, 11.86557637584724, 12.39738185863570, 12.83811092196647, 13.71416630774887, 14.04433682819312, 14.45614278696291, 15.26169834254907

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