Properties

Label 2-30960-1.1-c1-0-28
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 − 4·7-s + 4·11-s + 2·13-s − 6·17-s + 6·19-s − 6·23-s + 25-s + 2·29-s + 4·31-s + 4·35-s + 8·37-s − 8·41-s + 43-s − 6·47-s + 9·49-s − 6·53-s − 4·55-s − 10·61-s − 2·65-s − 12·67-s + 16·71-s + 16·73-s − 16·77-s + 4·79-s + 6·83-s + 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s + 1.31·37-s − 1.24·41-s + 0.152·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 1.28·61-s − 0.248·65-s − 1.46·67-s + 1.89·71-s + 1.87·73-s − 1.82·77-s + 0.450·79-s + 0.658·83-s + 0.650·85-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−15.48849155552248, −15.00617997735590, −14.12304068803770, −13.72673538404898, −13.37482013578134, −12.66544558071299, −12.13793945461073, −11.72027893520235, −11.19571426341070, −10.54032288676623, −9.842558821266034, −9.348979524811985, −9.138974560439199, −8.191454702292971, −7.849837511010118, −6.798006081941379, −6.575282281765516, −6.207540254966218, −5.351162225792665, −4.446324637975131, −3.981556383801455, −3.346675728140388, −2.834437690352454, −1.829447984831378, −0.9044209494024330, 0, 0.9044209494024330, 1.829447984831378, 2.834437690352454, 3.346675728140388, 3.981556383801455, 4.446324637975131, 5.351162225792665, 6.207540254966218, 6.575282281765516, 6.798006081941379, 7.849837511010118, 8.191454702292971, 9.138974560439199, 9.348979524811985, 9.842558821266034, 10.54032288676623, 11.19571426341070, 11.72027893520235, 12.13793945461073, 12.66544558071299, 13.37482013578134, 13.72673538404898, 14.12304068803770, 15.00617997735590, 15.48849155552248

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