Properties

Label 2-30960-1.1-c1-0-36
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 + 2·7-s − 2·11-s + 2·13-s + 8·19-s − 8·23-s + 25-s − 2·29-s − 2·35-s − 8·37-s + 8·41-s + 43-s − 3·49-s − 2·53-s + 2·55-s + 6·59-s + 8·61-s − 2·65-s − 12·67-s − 2·73-s − 4·77-s + 12·79-s − 18·83-s − 6·89-s + 4·91-s − 8·95-s − 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 0.603·11-s + 0.554·13-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.338·35-s − 1.31·37-s + 1.24·41-s + 0.152·43-s − 3/7·49-s − 0.274·53-s + 0.269·55-s + 0.781·59-s + 1.02·61-s − 0.248·65-s − 1.46·67-s − 0.234·73-s − 0.455·77-s + 1.35·79-s − 1.97·83-s − 0.635·89-s + 0.419·91-s − 0.820·95-s − 0.203·97-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
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

−15.59021814025869, −14.71412387677861, −14.30631433479854, −13.84569777000078, −13.34131239069849, −12.67537368968635, −12.03348279632258, −11.68499330691281, −11.15794481294379, −10.62175960067797, −9.988708530563534, −9.505318002102385, −8.731813473304244, −8.233460897561101, −7.682700312507934, −7.366206502942887, −6.556239727711692, −5.699647093566488, −5.413335654663734, −4.658213156648561, −3.982837057338385, −3.413361110882083, −2.641815405691415, −1.792379043987789, −1.073338809457541, 0, 1.073338809457541, 1.792379043987789, 2.641815405691415, 3.413361110882083, 3.982837057338385, 4.658213156648561, 5.413335654663734, 5.699647093566488, 6.556239727711692, 7.366206502942887, 7.682700312507934, 8.233460897561101, 8.731813473304244, 9.505318002102385, 9.988708530563534, 10.62175960067797, 11.15794481294379, 11.68499330691281, 12.03348279632258, 12.67537368968635, 13.34131239069849, 13.84569777000078, 14.30631433479854, 14.71412387677861, 15.59021814025869

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