Properties

Label 2-30960-1.1-c1-0-34
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 − 6·11-s + 2·13-s + 4·17-s − 2·19-s + 8·23-s + 25-s − 6·29-s + 8·37-s + 2·41-s − 43-s − 8·47-s − 7·49-s − 4·53-s − 6·55-s − 10·59-s + 10·61-s + 2·65-s − 4·67-s + 4·71-s + 8·79-s − 8·83-s + 4·85-s − 2·89-s − 2·95-s − 18·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.80·11-s + 0.554·13-s + 0.970·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 1.31·37-s + 0.312·41-s − 0.152·43-s − 1.16·47-s − 49-s − 0.549·53-s − 0.809·55-s − 1.30·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s + 0.474·71-s + 0.900·79-s − 0.878·83-s + 0.433·85-s − 0.211·89-s − 0.205·95-s − 1.82·97-s + 0.0995·101-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−15.24165084973893, −14.90212575538794, −14.39286067720653, −13.58729407647826, −13.23571109151968, −12.77842398623306, −12.47352228337272, −11.41027007297832, −11.01661314250556, −10.69654163000159, −9.827588049994861, −9.660005185191667, −8.836280684432384, −8.216938259832799, −7.761115629041556, −7.229292833583535, −6.427399211899068, −5.889695286817866, −5.190945191806498, −4.946678956205754, −3.989058099569767, −3.107051875822955, −2.772732110571319, −1.867975874968720, −1.057014648951379, 0, 1.057014648951379, 1.867975874968720, 2.772732110571319, 3.107051875822955, 3.989058099569767, 4.946678956205754, 5.190945191806498, 5.889695286817866, 6.427399211899068, 7.229292833583535, 7.761115629041556, 8.216938259832799, 8.836280684432384, 9.660005185191667, 9.827588049994861, 10.69654163000159, 11.01661314250556, 11.41027007297832, 12.47352228337272, 12.77842398623306, 13.23571109151968, 13.58729407647826, 14.39286067720653, 14.90212575538794, 15.24165084973893

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