Properties

Label 2-23520-1.1-c1-0-26
Degree $2$
Conductor $23520$
Sign $-1$
Analytic cond. $187.808$
Root an. cond. $13.7043$
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  − 3-s + 5-s + 9-s − 4·11-s − 6·13-s − 15-s − 2·17-s + 4·19-s + 25-s − 27-s + 10·29-s − 4·31-s + 4·33-s − 10·37-s + 6·39-s − 2·41-s + 4·43-s + 45-s + 8·47-s + 2·51-s + 2·53-s − 4·55-s − 4·57-s + 12·59-s + 10·61-s − 6·65-s − 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.696·33-s − 1.64·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.744·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23520\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(187.808\)
Root analytic conductor: \(13.7043\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{23520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23520,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.84970496366102, −15.29929202400703, −14.54777682432110, −14.16217558177183, −13.45144214629087, −13.06918703474457, −12.31370529150306, −12.06195915237754, −11.45495522751180, −10.56607029164118, −10.30157042941417, −9.910230624029351, −9.115093370091418, −8.585109825970478, −7.738754301185515, −7.236928617726677, −6.829709607102357, −5.959289094311404, −5.282012359226417, −5.037464587145213, −4.364732871695893, −3.331601065101748, −2.556948538237853, −2.090765018898482, −0.9109774686089874, 0, 0.9109774686089874, 2.090765018898482, 2.556948538237853, 3.331601065101748, 4.364732871695893, 5.037464587145213, 5.282012359226417, 5.959289094311404, 6.829709607102357, 7.236928617726677, 7.738754301185515, 8.585109825970478, 9.115093370091418, 9.910230624029351, 10.30157042941417, 10.56607029164118, 11.45495522751180, 12.06195915237754, 12.31370529150306, 13.06918703474457, 13.45144214629087, 14.16217558177183, 14.54777682432110, 15.29929202400703, 15.84970496366102

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