Properties

Label 2-23520-1.1-c1-0-7
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 $0$

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 + 6·17-s + 4·19-s + 8·23-s + 25-s + 27-s + 10·29-s + 4·31-s − 4·33-s − 6·37-s − 6·39-s − 6·41-s − 4·43-s − 45-s − 12·47-s + 6·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s + 2·61-s + 6·65-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 + 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.718·31-s − 0.696·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.744·65-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: \(0\)
Selberg data: \((2,\ 23520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.162871107\)
\(L(\frac12)\) \(\approx\) \(2.162871107\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.33485740989688, −14.94532276473920, −14.43366378413432, −13.88765088791533, −13.32345974703862, −12.73493552234680, −12.11483407339370, −11.91648218207932, −11.07173962135358, −10.26267791934422, −9.988875208882367, −9.582684534095771, −8.503504510350801, −8.329297905271932, −7.565341686663925, −7.187673857018493, −6.661856274628163, −5.497565361462662, −4.957997292083325, −4.760297125575112, −3.540156533823400, −2.935290006531299, −2.682667259198023, −1.499566579414323, −0.5816266994947348, 0.5816266994947348, 1.499566579414323, 2.682667259198023, 2.935290006531299, 3.540156533823400, 4.760297125575112, 4.957997292083325, 5.497565361462662, 6.661856274628163, 7.187673857018493, 7.565341686663925, 8.329297905271932, 8.503504510350801, 9.582684534095771, 9.988875208882367, 10.26267791934422, 11.07173962135358, 11.91648218207932, 12.11483407339370, 12.73493552234680, 13.32345974703862, 13.88765088791533, 14.43366378413432, 14.94532276473920, 15.33485740989688

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