Properties

Label 2-30960-1.1-c1-0-30
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 4·11-s − 13-s − 19-s − 4·23-s + 25-s + 5·29-s + 9·31-s − 35-s + 4·37-s + 7·41-s + 43-s + 6·47-s − 6·49-s + 2·53-s − 4·55-s − 7·61-s − 65-s − 15·67-s − 6·71-s − 5·73-s + 4·77-s − 9·79-s + 91-s − 95-s − 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.20·11-s − 0.277·13-s − 0.229·19-s − 0.834·23-s + 1/5·25-s + 0.928·29-s + 1.61·31-s − 0.169·35-s + 0.657·37-s + 1.09·41-s + 0.152·43-s + 0.875·47-s − 6/7·49-s + 0.274·53-s − 0.539·55-s − 0.896·61-s − 0.124·65-s − 1.83·67-s − 0.712·71-s − 0.585·73-s + 0.455·77-s − 1.01·79-s + 0.104·91-s − 0.102·95-s − 0.203·97-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: \(1\)
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 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.40086304952135, −14.87938384545544, −14.20747866555536, −13.72861641841932, −13.30699149628607, −12.74406963661521, −12.23999775213541, −11.74907644860998, −10.95924309670732, −10.46245112815196, −9.989483194923194, −9.630055473868570, −8.786875138072462, −8.345072302608959, −7.646950877855879, −7.260177179478469, −6.251073356459129, −6.095078904729772, −5.351809463951781, −4.568069627370976, −4.237646331319887, −2.987405232186586, −2.780839209868984, −1.976498286205136, −0.9721847765923055, 0, 0.9721847765923055, 1.976498286205136, 2.780839209868984, 2.987405232186586, 4.237646331319887, 4.568069627370976, 5.351809463951781, 6.095078904729772, 6.251073356459129, 7.260177179478469, 7.646950877855879, 8.345072302608959, 8.786875138072462, 9.630055473868570, 9.989483194923194, 10.46245112815196, 10.95924309670732, 11.74907644860998, 12.23999775213541, 12.74406963661521, 13.30699149628607, 13.72861641841932, 14.20747866555536, 14.87938384545544, 15.40086304952135

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