Properties

Label 2-30960-1.1-c1-0-24
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 − 5·11-s − 3·13-s + 3·17-s − 4·19-s − 7·23-s + 25-s + 8·29-s + 3·31-s − 3·41-s − 43-s − 7·49-s + 9·53-s + 5·55-s + 8·59-s + 3·65-s + 11·67-s + 8·71-s − 4·73-s + 8·79-s + 9·83-s − 3·85-s + 4·89-s + 4·95-s + 13·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s − 0.832·13-s + 0.727·17-s − 0.917·19-s − 1.45·23-s + 1/5·25-s + 1.48·29-s + 0.538·31-s − 0.468·41-s − 0.152·43-s − 49-s + 1.23·53-s + 0.674·55-s + 1.04·59-s + 0.372·65-s + 1.34·67-s + 0.949·71-s − 0.468·73-s + 0.900·79-s + 0.987·83-s − 0.325·85-s + 0.423·89-s + 0.410·95-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.50048409900971, −14.76284075426217, −14.43965909098539, −13.73678731303046, −13.25381692240321, −12.64767420972625, −12.12302387505459, −11.86286289152017, −11.00468699960580, −10.48308959573782, −9.984780282496603, −9.725788156731119, −8.556353839017263, −8.301369983700698, −7.794091664109969, −7.241135651934334, −6.505779930120801, −5.956946416097138, −5.062196204582516, −4.870644435412997, −3.993372397131550, −3.342217128712832, −2.488209683809479, −2.135760207753469, −0.8252989054761261, 0, 0.8252989054761261, 2.135760207753469, 2.488209683809479, 3.342217128712832, 3.993372397131550, 4.870644435412997, 5.062196204582516, 5.956946416097138, 6.505779930120801, 7.241135651934334, 7.794091664109969, 8.301369983700698, 8.556353839017263, 9.725788156731119, 9.984780282496603, 10.48308959573782, 11.00468699960580, 11.86286289152017, 12.12302387505459, 12.64767420972625, 13.25381692240321, 13.73678731303046, 14.43965909098539, 14.76284075426217, 15.50048409900971

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