Properties

Label 2-31200-1.1-c1-0-52
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
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 + 3·7-s + 9-s + 5·11-s + 13-s − 5·17-s + 4·19-s − 3·21-s + 2·23-s − 27-s − 9·29-s + 3·31-s − 5·33-s − 10·37-s − 39-s − 12·41-s + 2·43-s + 9·47-s + 2·49-s + 5·51-s + 9·53-s − 4·57-s + 3·59-s − 7·61-s + 3·63-s + 9·67-s − 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 1.21·17-s + 0.917·19-s − 0.654·21-s + 0.417·23-s − 0.192·27-s − 1.67·29-s + 0.538·31-s − 0.870·33-s − 1.64·37-s − 0.160·39-s − 1.87·41-s + 0.304·43-s + 1.31·47-s + 2/7·49-s + 0.700·51-s + 1.23·53-s − 0.529·57-s + 0.390·59-s − 0.896·61-s + 0.377·63-s + 1.09·67-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.25340151992276, −14.95291312645342, −14.25577774964926, −13.74013449003264, −13.41852280525357, −12.56674513154296, −11.99035947765975, −11.54706746788939, −11.28682627469917, −10.67708375976505, −10.07309733889844, −9.327332003791844, −8.813985492511090, −8.481889548400698, −7.543219059480366, −7.042689729910114, −6.634710888422937, −5.828097210563688, −5.276955346314949, −4.743966174348442, −3.983579115123844, −3.624356456273572, −2.464520171943432, −1.575990606656140, −1.248702703744989, 0, 1.248702703744989, 1.575990606656140, 2.464520171943432, 3.624356456273572, 3.983579115123844, 4.743966174348442, 5.276955346314949, 5.828097210563688, 6.634710888422937, 7.042689729910114, 7.543219059480366, 8.481889548400698, 8.813985492511090, 9.327332003791844, 10.07309733889844, 10.67708375976505, 11.28682627469917, 11.54706746788939, 11.99035947765975, 12.56674513154296, 13.41852280525357, 13.74013449003264, 14.25577774964926, 14.95291312645342, 15.25340151992276

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