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Properties

Label 2-369600-1.1-c1-0-135
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 + 7-s + 9-s + 11-s + 2·13-s + 6·17-s − 2·19-s + 21-s + 27-s − 8·29-s + 8·31-s + 33-s − 6·37-s + 2·39-s − 8·41-s − 8·43-s − 10·47-s + 49-s + 6·51-s − 2·53-s − 2·57-s − 8·59-s − 10·61-s + 63-s − 2·67-s + 10·73-s + 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 0.218·21-s + 0.192·27-s − 1.48·29-s + 1.43·31-s + 0.174·33-s − 0.986·37-s + 0.320·39-s − 1.24·41-s − 1.21·43-s − 1.45·47-s + 1/7·49-s + 0.840·51-s − 0.274·53-s − 0.264·57-s − 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.244·67-s + 1.17·73-s + 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.478363569\)
\(L(\frac12)\) \(\approx\) \(3.478363569\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−12.46861304200091, −11.96448353013033, −11.73905368330740, −11.18671348984130, −10.48866499670676, −10.35189726796641, −9.771325203277918, −9.228450033553155, −8.943108004281540, −8.266007727334628, −7.958658176240937, −7.684447727050630, −6.972285563196787, −6.431838633473328, −6.163230731670793, −5.403925193238287, −4.927694490535551, −4.597945663224381, −3.683118178761566, −3.476764720127784, −3.096455132826063, −2.181883515884719, −1.703417066200931, −1.294261766416760, −0.4566645591738659, 0.4566645591738659, 1.294261766416760, 1.703417066200931, 2.181883515884719, 3.096455132826063, 3.476764720127784, 3.683118178761566, 4.597945663224381, 4.927694490535551, 5.403925193238287, 6.163230731670793, 6.431838633473328, 6.972285563196787, 7.684447727050630, 7.958658176240937, 8.266007727334628, 8.943108004281540, 9.228450033553155, 9.771325203277918, 10.35189726796641, 10.48866499670676, 11.18671348984130, 11.73905368330740, 11.96448353013033, 12.46861304200091

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