Properties

Label 2-31200-1.1-c1-0-30
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 + 2·7-s + 9-s − 2·11-s + 13-s + 2·17-s − 2·19-s − 2·21-s − 8·23-s − 27-s − 6·29-s + 2·31-s + 2·33-s − 2·37-s − 39-s − 2·41-s + 6·47-s − 3·49-s − 2·51-s + 10·53-s + 2·57-s + 14·59-s − 10·61-s + 2·63-s + 2·67-s + 8·69-s + 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 0.458·19-s − 0.436·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.348·33-s − 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s + 0.264·57-s + 1.82·59-s − 1.28·61-s + 0.251·63-s + 0.244·67-s + 0.963·69-s + 0.712·71-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.37419596273581, −14.89419776717176, −14.25636680140938, −13.79560028570645, −13.23539577682769, −12.62990420198145, −12.12180774872959, −11.60834395605934, −11.14911446394977, −10.53341125738756, −10.11611679071862, −9.589564711506820, −8.667913421019187, −8.340787838243811, −7.584874274991877, −7.306850714116807, −6.324494891711121, −5.927012909778251, −5.281068638189191, −4.800287899297651, −3.989502185001076, −3.552932252527984, −2.374585454466704, −1.913166226123941, −0.9690795976854270, 0, 0.9690795976854270, 1.913166226123941, 2.374585454466704, 3.552932252527984, 3.989502185001076, 4.800287899297651, 5.281068638189191, 5.927012909778251, 6.324494891711121, 7.306850714116807, 7.584874274991877, 8.340787838243811, 8.667913421019187, 9.589564711506820, 10.11611679071862, 10.53341125738756, 11.14911446394977, 11.60834395605934, 12.12180774872959, 12.62990420198145, 13.23539577682769, 13.79560028570645, 14.25636680140938, 14.89419776717176, 15.37419596273581

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