Properties

Label 2-30960-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·13-s − 4·19-s + 8·23-s + 25-s − 2·29-s + 4·31-s − 2·37-s − 10·41-s + 43-s − 12·47-s − 7·49-s − 2·53-s + 8·59-s + 12·61-s + 4·65-s − 8·67-s + 12·71-s − 6·73-s + 6·83-s − 12·89-s + 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.10·13-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s + 0.152·43-s − 1.75·47-s − 49-s − 0.274·53-s + 1.04·59-s + 1.53·61-s + 0.496·65-s − 0.977·67-s + 1.42·71-s − 0.702·73-s + 0.658·83-s − 1.27·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: \(0\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.161484595\)
\(L(\frac12)\) \(\approx\) \(1.161484595\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.94413094260769, −14.79454572931577, −14.17064785049116, −13.33270826041692, −13.01899912985441, −12.47084521727495, −11.90557547045261, −11.34934892909881, −10.94758572080646, −10.17661056406368, −9.802691870072741, −9.151523393728315, −8.428188087561120, −8.164871798375093, −7.316781548790886, −6.839485902131922, −6.449432612878456, −5.418652353057397, −4.947201588751332, −4.471873374715023, −3.600132851280405, −3.027403590520520, −2.294581764982286, −1.472452439758246, −0.4104509558882428, 0.4104509558882428, 1.472452439758246, 2.294581764982286, 3.027403590520520, 3.600132851280405, 4.471873374715023, 4.947201588751332, 5.418652353057397, 6.449432612878456, 6.839485902131922, 7.316781548790886, 8.164871798375093, 8.428188087561120, 9.151523393728315, 9.802691870072741, 10.17661056406368, 10.94758572080646, 11.34934892909881, 11.90557547045261, 12.47084521727495, 13.01899912985441, 13.33270826041692, 14.17064785049116, 14.79454572931577, 14.94413094260769

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