Properties

Label 2-51600-1.1-c1-0-38
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 − 2·7-s + 9-s − 5·11-s + 5·13-s − 17-s + 2·19-s + 2·21-s + 23-s − 27-s + 8·29-s + 7·31-s + 5·33-s + 4·37-s − 5·39-s + 9·41-s + 43-s + 8·47-s − 3·49-s + 51-s + 5·53-s − 2·57-s + 4·59-s + 4·61-s − 2·63-s + 11·67-s − 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s − 0.192·27-s + 1.48·29-s + 1.25·31-s + 0.870·33-s + 0.657·37-s − 0.800·39-s + 1.40·41-s + 0.152·43-s + 1.16·47-s − 3/7·49-s + 0.140·51-s + 0.686·53-s − 0.264·57-s + 0.520·59-s + 0.512·61-s − 0.251·63-s + 1.34·67-s − 0.120·69-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

\(L(1)\) \(\approx\) \(2.014319053\)
\(L(\frac12)\) \(\approx\) \(2.014319053\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−14.33733859346421, −13.85538159318920, −13.38717001867149, −12.93080584043308, −12.56151410211134, −11.95304674748878, −11.25126031858968, −10.95968306821438, −10.34609981744473, −9.967556348057843, −9.414702234077682, −8.630776787159064, −8.234727192008934, −7.651648632328431, −6.981664727833977, −6.360952960447146, −6.035745492367891, −5.370837903678800, −4.840679449565464, −4.135288184223970, −3.500796172544208, −2.736194984213660, −2.319635753204605, −0.9989952995424458, −0.6497729768301859, 0.6497729768301859, 0.9989952995424458, 2.319635753204605, 2.736194984213660, 3.500796172544208, 4.135288184223970, 4.840679449565464, 5.370837903678800, 6.035745492367891, 6.360952960447146, 6.981664727833977, 7.651648632328431, 8.234727192008934, 8.630776787159064, 9.414702234077682, 9.967556348057843, 10.34609981744473, 10.95968306821438, 11.25126031858968, 11.95304674748878, 12.56151410211134, 12.93080584043308, 13.38717001867149, 13.85538159318920, 14.33733859346421

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