Properties

Label 2-187200-1.1-c1-0-106
Degree $2$
Conductor $187200$
Sign $1$
Analytic cond. $1494.79$
Root an. cond. $38.6626$
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  + 13-s + 6·17-s + 8·19-s − 6·23-s − 2·29-s + 6·31-s − 2·37-s − 10·41-s − 4·43-s + 4·47-s − 7·49-s + 4·53-s − 4·59-s + 2·61-s − 10·67-s + 12·73-s + 4·83-s − 14·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.277·13-s + 1.45·17-s + 1.83·19-s − 1.25·23-s − 0.371·29-s + 1.07·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s − 49-s + 0.549·53-s − 0.520·59-s + 0.256·61-s − 1.22·67-s + 1.40·73-s + 0.439·83-s − 1.48·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187200\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1494.79\)
Root analytic conductor: \(38.6626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 187200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.413889408\)
\(L(\frac12)\) \(\approx\) \(2.413889408\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.35914454317467, −12.46526304029497, −12.06379376288141, −11.89190603548357, −11.35823014026772, −10.73715843981834, −10.13288424417387, −9.851368340210790, −9.532421275973703, −8.792787662232087, −8.278654921500515, −7.826515632409042, −7.494638842399321, −6.846208958440747, −6.305389076915281, −5.793289667784531, −5.171652480415210, −5.030633209804864, −4.025751828466724, −3.661135674513543, −3.087180765922624, −2.615070955922700, −1.597523438977111, −1.325237288483687, −0.4560655539652381, 0.4560655539652381, 1.325237288483687, 1.597523438977111, 2.615070955922700, 3.087180765922624, 3.661135674513543, 4.025751828466724, 5.030633209804864, 5.171652480415210, 5.793289667784531, 6.305389076915281, 6.846208958440747, 7.494638842399321, 7.826515632409042, 8.278654921500515, 8.792787662232087, 9.532421275973703, 9.851368340210790, 10.13288424417387, 10.73715843981834, 11.35823014026772, 11.89190603548357, 12.06379376288141, 12.46526304029497, 13.35914454317467

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