Properties

Label 2-51600-1.1-c1-0-39
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
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 + 4·7-s + 9-s − 6·13-s + 6·17-s + 4·19-s + 4·21-s + 27-s + 2·29-s − 8·31-s − 10·37-s − 6·39-s − 6·41-s − 43-s + 8·47-s + 9·49-s + 6·51-s + 6·53-s + 4·57-s + 10·61-s + 4·63-s − 4·67-s − 8·71-s + 6·73-s + 81-s + 16·83-s + 2·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 1.64·37-s − 0.960·39-s − 0.937·41-s − 0.152·43-s + 1.16·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.488·67-s − 0.949·71-s + 0.702·73-s + 1/9·81-s + 1.75·83-s + 0.214·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.782757516\)
\(L(\frac12)\) \(\approx\) \(3.782757516\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.54665534800345, −14.05139295577399, −13.72536045034262, −12.96232230463950, −12.20774177011218, −12.00502622684652, −11.64456738035870, −10.63972624341195, −10.48452219300279, −9.751495680713697, −9.324492122024027, −8.675180896059776, −8.122866670092710, −7.637938333028000, −7.281548604756946, −6.801728580033069, −5.538527270662368, −5.324146452136019, −4.888085651964377, −4.088856894814249, −3.461959020695263, −2.789391670000864, −2.024051158375667, −1.560410552816762, −0.6590617167263241, 0.6590617167263241, 1.560410552816762, 2.024051158375667, 2.789391670000864, 3.461959020695263, 4.088856894814249, 4.888085651964377, 5.324146452136019, 5.538527270662368, 6.801728580033069, 7.281548604756946, 7.637938333028000, 8.122866670092710, 8.675180896059776, 9.324492122024027, 9.751495680713697, 10.48452219300279, 10.63972624341195, 11.64456738035870, 12.00502622684652, 12.20774177011218, 12.96232230463950, 13.72536045034262, 14.05139295577399, 14.54665534800345

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