Properties

Label 2-30960-1.1-c1-0-27
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 − 2·13-s − 2·17-s − 4·19-s + 25-s − 2·29-s − 4·35-s + 10·37-s + 6·41-s + 43-s − 8·47-s + 9·49-s − 2·53-s + 2·61-s − 2·65-s + 4·67-s + 8·71-s + 10·73-s + 8·79-s + 16·83-s − 2·85-s − 6·89-s + 8·91-s − 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.676·35-s + 1.64·37-s + 0.937·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s + 0.256·61-s − 0.248·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.900·79-s + 1.75·83-s − 0.216·85-s − 0.635·89-s + 0.838·91-s − 0.410·95-s + 0.203·97-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: $\chi_{30960} (1, \cdot )$
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.34243011534808, −14.85158476972914, −14.34125226914811, −13.68436865778589, −13.03377609124997, −12.90592245854531, −12.37999593012863, −11.66302634506956, −10.97935304287751, −10.54643815253896, −9.835374624574290, −9.437562697694741, −9.162542692658279, −8.259026313018303, −7.739794842145171, −6.930473308086729, −6.442477897400823, −6.139734155663259, −5.357563577206139, −4.648855291839126, −3.954290328859558, −3.316835708705932, −2.525033236663605, −2.120041392965691, −0.8700518633808493, 0, 0.8700518633808493, 2.120041392965691, 2.525033236663605, 3.316835708705932, 3.954290328859558, 4.648855291839126, 5.357563577206139, 6.139734155663259, 6.442477897400823, 6.930473308086729, 7.739794842145171, 8.259026313018303, 9.162542692658279, 9.437562697694741, 9.835374624574290, 10.54643815253896, 10.97935304287751, 11.66302634506956, 12.37999593012863, 12.90592245854531, 13.03377609124997, 13.68436865778589, 14.34125226914811, 14.85158476972914, 15.34243011534808

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