Properties

Label 2-31200-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 13-s − 2·17-s − 4·19-s − 4·23-s + 27-s − 6·29-s + 8·31-s + 6·37-s + 39-s − 2·41-s − 4·43-s − 7·49-s − 2·51-s − 6·53-s − 4·57-s − 2·61-s − 8·67-s − 4·69-s − 6·73-s − 4·79-s + 81-s + 12·83-s − 6·87-s + 6·89-s + 8·93-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s − 49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.256·61-s − 0.977·67-s − 0.481·69-s − 0.702·73-s − 0.450·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 0.635·89-s + 0.829·93-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: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.257273387\)
\(L(\frac12)\) \(\approx\) \(2.257273387\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.98678852198887, −14.66941497366746, −14.00361011370497, −13.49865103327871, −13.05370680185735, −12.59876935831881, −11.83950109095164, −11.41211693718539, −10.78259193902851, −10.16942698290602, −9.742255694590774, −9.048664902665455, −8.605962985000364, −7.985742762117786, −7.605679098855739, −6.759207935749016, −6.255957950913210, −5.768596413522466, −4.650476871151049, −4.478624530517933, −3.582482799966235, −3.040713025172574, −2.156852663763661, −1.690200121551763, −0.5354784637577695, 0.5354784637577695, 1.690200121551763, 2.156852663763661, 3.040713025172574, 3.582482799966235, 4.478624530517933, 4.650476871151049, 5.768596413522466, 6.255957950913210, 6.759207935749016, 7.605679098855739, 7.985742762117786, 8.605962985000364, 9.048664902665455, 9.742255694590774, 10.16942698290602, 10.78259193902851, 11.41211693718539, 11.83950109095164, 12.59876935831881, 13.05370680185735, 13.49865103327871, 14.00361011370497, 14.66941497366746, 14.98678852198887

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