Properties

Label 2-30960-1.1-c1-0-53
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 4·11-s − 6·13-s − 2·17-s + 6·23-s + 25-s + 2·29-s − 4·31-s + 2·35-s − 6·41-s − 43-s − 10·47-s − 3·49-s + 4·55-s − 4·59-s − 12·61-s + 6·65-s + 4·67-s − 2·73-s + 8·77-s − 16·79-s − 12·83-s + 2·85-s + 6·89-s + 12·91-s − 18·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.338·35-s − 0.937·41-s − 0.152·43-s − 1.45·47-s − 3/7·49-s + 0.539·55-s − 0.520·59-s − 1.53·61-s + 0.744·65-s + 0.488·67-s − 0.234·73-s + 0.911·77-s − 1.80·79-s − 1.31·83-s + 0.216·85-s + 0.635·89-s + 1.25·91-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.60251079523023, −15.16007854846911, −14.66505375314465, −14.12127834427666, −13.24409762572177, −13.03771309458185, −12.51991258202433, −12.01301431073723, −11.31951845046360, −10.85853237569532, −10.13008761373904, −9.859842564860814, −9.171836094963031, −8.595108051989054, −7.912422879638209, −7.369630730782454, −6.952174902110625, −6.335157040529837, −5.468141692953639, −4.904078281398964, −4.560414163155096, −3.508469315337038, −2.926375582135306, −2.487197735681379, −1.471027809636395, 0, 0, 1.471027809636395, 2.487197735681379, 2.926375582135306, 3.508469315337038, 4.560414163155096, 4.904078281398964, 5.468141692953639, 6.335157040529837, 6.952174902110625, 7.369630730782454, 7.912422879638209, 8.595108051989054, 9.171836094963031, 9.859842564860814, 10.13008761373904, 10.85853237569532, 11.31951845046360, 12.01301431073723, 12.51991258202433, 13.03771309458185, 13.24409762572177, 14.12127834427666, 14.66505375314465, 15.16007854846911, 15.60251079523023

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