Properties

Label 2-30960-1.1-c1-0-49
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 + 4·13-s − 4·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s + 4·31-s + 4·35-s + 2·37-s − 10·41-s + 43-s + 4·47-s + 9·49-s + 2·53-s − 4·55-s − 12·59-s + 4·65-s − 8·67-s − 12·71-s − 14·73-s − 16·77-s − 8·79-s + 6·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 1.20·11-s + 1.10·13-s − 0.970·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s − 1.56·41-s + 0.152·43-s + 0.583·47-s + 9/7·49-s + 0.274·53-s − 0.539·55-s − 1.56·59-s + 0.496·65-s − 0.977·67-s − 1.42·71-s − 1.63·73-s − 1.82·77-s − 0.900·79-s + 0.658·83-s − 0.433·85-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−15.38051325301860, −14.82711768633896, −14.44103250115402, −13.52451174666502, −13.30819925287600, −13.04160471982081, −12.08567165085941, −11.52368901145783, −10.97265485238109, −10.63881458531797, −10.30503455742277, −9.093592006945853, −8.963261274437155, −8.264187848849934, −7.856766182821188, −7.139938141099991, −6.548239288904613, −5.758455459159680, −5.339942111119706, −4.594983342024729, −4.310604716839754, −3.196199217105689, −2.555422519113520, −1.776843263210540, −1.255727938678685, 0, 1.255727938678685, 1.776843263210540, 2.555422519113520, 3.196199217105689, 4.310604716839754, 4.594983342024729, 5.339942111119706, 5.758455459159680, 6.548239288904613, 7.139938141099991, 7.856766182821188, 8.264187848849934, 8.963261274437155, 9.093592006945853, 10.30503455742277, 10.63881458531797, 10.97265485238109, 11.52368901145783, 12.08567165085941, 13.04160471982081, 13.30819925287600, 13.52451174666502, 14.44103250115402, 14.82711768633896, 15.38051325301860

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