Properties

Label 2-30960-1.1-c1-0-1
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 − 7-s − 5·13-s − 4·17-s + 3·19-s + 25-s − 7·29-s − 7·31-s − 35-s − 8·37-s − 41-s + 43-s − 6·47-s − 6·49-s + 2·53-s + 5·61-s − 5·65-s − 11·67-s + 6·71-s − 5·73-s − 79-s − 8·83-s − 4·85-s + 12·89-s + 5·91-s + 3·95-s + 18·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.38·13-s − 0.970·17-s + 0.688·19-s + 1/5·25-s − 1.29·29-s − 1.25·31-s − 0.169·35-s − 1.31·37-s − 0.156·41-s + 0.152·43-s − 0.875·47-s − 6/7·49-s + 0.274·53-s + 0.640·61-s − 0.620·65-s − 1.34·67-s + 0.712·71-s − 0.585·73-s − 0.112·79-s − 0.878·83-s − 0.433·85-s + 1.27·89-s + 0.524·91-s + 0.307·95-s + 1.82·97-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

\(L(1)\) \(\approx\) \(0.9569143530\)
\(L(\frac12)\) \(\approx\) \(0.9569143530\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−14.97553777016138, −14.59270305831251, −14.11550102847840, −13.43708993425826, −12.90295626148971, −12.68697734961176, −11.75507319200859, −11.55268434130141, −10.71534361410822, −10.24007489701872, −9.689650088673883, −9.140634996741597, −8.842274352641480, −7.848869357447233, −7.378665644293186, −6.868855838165052, −6.284013691653720, −5.462392434734084, −5.126233022442936, −4.408042059182416, −3.580383905410117, −3.001911545728620, −2.129993035330794, −1.694125595093256, −0.3502303500500625, 0.3502303500500625, 1.694125595093256, 2.129993035330794, 3.001911545728620, 3.580383905410117, 4.408042059182416, 5.126233022442936, 5.462392434734084, 6.284013691653720, 6.868855838165052, 7.378665644293186, 7.848869357447233, 8.842274352641480, 9.140634996741597, 9.689650088673883, 10.24007489701872, 10.71534361410822, 11.55268434130141, 11.75507319200859, 12.68697734961176, 12.90295626148971, 13.43708993425826, 14.11550102847840, 14.59270305831251, 14.97553777016138

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