Properties

Label 2-51600-1.1-c1-0-22
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 + 7-s + 9-s − 5·11-s + 7·13-s − 4·17-s + 19-s + 21-s − 4·23-s + 27-s − 5·29-s + 10·31-s − 5·33-s − 10·37-s + 7·39-s + 43-s − 47-s − 6·49-s − 4·51-s − 12·53-s + 57-s − 4·59-s − 8·61-s + 63-s − 2·67-s − 4·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.94·13-s − 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 0.928·29-s + 1.79·31-s − 0.870·33-s − 1.64·37-s + 1.12·39-s + 0.152·43-s − 0.145·47-s − 6/7·49-s − 0.560·51-s − 1.64·53-s + 0.132·57-s − 0.520·59-s − 1.02·61-s + 0.125·63-s − 0.244·67-s − 0.481·69-s + 1.42·71-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

\(L(1)\) \(\approx\) \(2.402567611\)
\(L(\frac12)\) \(\approx\) \(2.402567611\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.27224796584726, −13.93391271114618, −13.51152636424649, −13.06347583509888, −12.66515088476162, −11.85181194849090, −11.33641851412156, −10.80071084182077, −10.46934295175333, −9.851239331308887, −9.193576959563405, −8.559487176447225, −8.281538326711085, −7.810115023474027, −7.212733489229992, −6.361372155794430, −6.077888283395585, −5.279324594411189, −4.678140049591469, −4.137519650699225, −3.308304457092630, −2.983375940748152, −1.974042809320886, −1.627805314691293, −0.5050441161436889, 0.5050441161436889, 1.627805314691293, 1.974042809320886, 2.983375940748152, 3.308304457092630, 4.137519650699225, 4.678140049591469, 5.279324594411189, 6.077888283395585, 6.361372155794430, 7.212733489229992, 7.810115023474027, 8.281538326711085, 8.559487176447225, 9.193576959563405, 9.851239331308887, 10.46934295175333, 10.80071084182077, 11.33641851412156, 11.85181194849090, 12.66515088476162, 13.06347583509888, 13.51152636424649, 13.93391271114618, 14.27224796584726

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