Properties

Label 2-23520-1.1-c1-0-24
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 − 5·11-s + 13-s + 15-s − 8·17-s + 5·19-s + 3·23-s + 25-s − 27-s + 4·29-s + 6·31-s + 5·33-s + 3·37-s − 39-s − 7·41-s − 8·43-s − 45-s − 47-s + 8·51-s + 9·53-s + 5·55-s − 5·57-s − 12·59-s + 4·61-s − 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 0.258·15-s − 1.94·17-s + 1.14·19-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 1.07·31-s + 0.870·33-s + 0.493·37-s − 0.160·39-s − 1.09·41-s − 1.21·43-s − 0.149·45-s − 0.145·47-s + 1.12·51-s + 1.23·53-s + 0.674·55-s − 0.662·57-s − 1.56·59-s + 0.512·61-s − 0.124·65-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: \(1\)
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.72212189493044, −15.30619445751557, −14.95221909123144, −13.80266870633096, −13.51440652476097, −13.13435059610014, −12.42417359790406, −11.84820957416336, −11.32739072097869, −10.92216508873981, −10.25843851167837, −9.909996141490678, −8.949823146460043, −8.527423723978058, −7.877598726240052, −7.275115203817708, −6.706295403695128, −6.133536762675841, −5.320188381438081, −4.787715358991309, −4.405553414908102, −3.327867209478664, −2.787883524454493, −1.940691876204550, −0.8482140674551937, 0, 0.8482140674551937, 1.940691876204550, 2.787883524454493, 3.327867209478664, 4.405553414908102, 4.787715358991309, 5.320188381438081, 6.133536762675841, 6.706295403695128, 7.275115203817708, 7.877598726240052, 8.527423723978058, 8.949823146460043, 9.909996141490678, 10.25843851167837, 10.92216508873981, 11.32739072097869, 11.84820957416336, 12.42417359790406, 13.13435059610014, 13.51440652476097, 13.80266870633096, 14.95221909123144, 15.30619445751557, 15.72212189493044

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