Properties

Label 2-30960-1.1-c1-0-32
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 + 4·17-s + 25-s − 2·29-s + 8·31-s − 2·35-s + 8·37-s − 8·41-s + 43-s − 8·47-s − 3·49-s + 14·53-s + 2·55-s − 10·59-s − 12·61-s + 2·65-s + 12·67-s − 8·71-s − 2·73-s − 4·77-s − 4·79-s − 6·83-s − 4·85-s − 18·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s − 0.603·11-s − 0.554·13-s + 0.970·17-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.338·35-s + 1.31·37-s − 1.24·41-s + 0.152·43-s − 1.16·47-s − 3/7·49-s + 1.92·53-s + 0.269·55-s − 1.30·59-s − 1.53·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.455·77-s − 0.450·79-s − 0.658·83-s − 0.433·85-s − 1.90·89-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 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.09478956757435, −15.04179696964534, −14.37253537710135, −13.77715190586967, −13.32753039546028, −12.60038651357414, −12.16407845045356, −11.62735038369910, −11.20289118732878, −10.54885394450744, −9.957875031110963, −9.609140033376561, −8.691575407271510, −8.160440127252094, −7.833689333639624, −7.247560258368363, −6.599840730499039, −5.792453098427646, −5.270605934604813, −4.620997501828107, −4.187526394772260, −3.182520110513976, −2.758168130847709, −1.808691097423709, −1.040362014834555, 0, 1.040362014834555, 1.808691097423709, 2.758168130847709, 3.182520110513976, 4.187526394772260, 4.620997501828107, 5.270605934604813, 5.792453098427646, 6.599840730499039, 7.247560258368363, 7.833689333639624, 8.160440127252094, 8.691575407271510, 9.609140033376561, 9.957875031110963, 10.54885394450744, 11.20289118732878, 11.62735038369910, 12.16407845045356, 12.60038651357414, 13.32753039546028, 13.77715190586967, 14.37253537710135, 15.04179696964534, 15.09478956757435

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