Properties

Label 2-23520-1.1-c1-0-12
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 + 2·11-s + 4·13-s + 15-s + 2·17-s + 6·19-s − 4·23-s + 25-s − 27-s − 10·29-s − 2·31-s − 2·33-s − 2·37-s − 4·39-s + 10·41-s + 4·43-s − 45-s + 8·47-s − 2·51-s + 4·53-s − 2·55-s − 6·57-s − 4·59-s + 2·61-s − 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s + 0.485·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s − 0.348·33-s − 0.328·37-s − 0.640·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 0.549·53-s − 0.269·55-s − 0.794·57-s − 0.520·59-s + 0.256·61-s − 0.496·65-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

\(L(1)\) \(\approx\) \(1.919327509\)
\(L(\frac12)\) \(\approx\) \(1.919327509\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−15.59354242520308, −14.96290779838822, −14.31528054142097, −13.87947212645485, −13.28484968060568, −12.61896739878891, −12.14769860707250, −11.62452304145245, −11.09179656919426, −10.77485358943510, −9.916160547951085, −9.382411500188184, −8.916787148825521, −8.113652515013280, −7.474656803691114, −7.176735366396268, −6.221095615013675, −5.774870613947657, −5.318989552486360, −4.335162985959343, −3.791871772976148, −3.356637509389883, −2.217775700338082, −1.318457445314737, −0.6407600344548730, 0.6407600344548730, 1.318457445314737, 2.217775700338082, 3.356637509389883, 3.791871772976148, 4.335162985959343, 5.318989552486360, 5.774870613947657, 6.221095615013675, 7.176735366396268, 7.474656803691114, 8.113652515013280, 8.916787148825521, 9.382411500188184, 9.916160547951085, 10.77485358943510, 11.09179656919426, 11.62452304145245, 12.14769860707250, 12.61896739878891, 13.28484968060568, 13.87947212645485, 14.31528054142097, 14.96290779838822, 15.59354242520308

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