Properties

Label 2-30960-1.1-c1-0-37
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 − 6·19-s − 6·23-s + 25-s − 6·29-s − 4·31-s − 4·35-s + 4·37-s + 43-s + 10·47-s + 9·49-s − 2·53-s + 4·59-s + 6·61-s − 2·65-s + 12·67-s − 4·73-s − 4·79-s + 6·83-s + 2·85-s − 10·89-s + 8·91-s + 6·95-s + 6·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 0.554·13-s − 0.485·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.676·35-s + 0.657·37-s + 0.152·43-s + 1.45·47-s + 9/7·49-s − 0.274·53-s + 0.520·59-s + 0.768·61-s − 0.248·65-s + 1.46·67-s − 0.468·73-s − 0.450·79-s + 0.658·83-s + 0.216·85-s − 1.05·89-s + 0.838·91-s + 0.615·95-s + 0.609·97-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 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.42399170154235, −14.71455699425413, −14.43341384698324, −13.92565184760068, −13.15268489574844, −12.79679600757915, −12.06887312082920, −11.57141332794157, −11.00181495306640, −10.85248854393033, −10.09311873169848, −9.333682430540891, −8.627665482539306, −8.381160137443558, −7.755288859645374, −7.286959697192675, −6.522042289299950, −5.836082835254377, −5.331352992355358, −4.471441322772623, −4.138824546235038, −3.563497564978531, −2.292073798208256, −2.016989466977614, −1.074192672917410, 0, 1.074192672917410, 2.016989466977614, 2.292073798208256, 3.563497564978531, 4.138824546235038, 4.471441322772623, 5.331352992355358, 5.836082835254377, 6.522042289299950, 7.286959697192675, 7.755288859645374, 8.381160137443558, 8.627665482539306, 9.333682430540891, 10.09311873169848, 10.85248854393033, 11.00181495306640, 11.57141332794157, 12.06887312082920, 12.79679600757915, 13.15268489574844, 13.92565184760068, 14.43341384698324, 14.71455699425413, 15.42399170154235

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