Properties

Label 2-51600-1.1-c1-0-30
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 + 9-s + 4·13-s − 4·19-s + 8·23-s − 27-s + 2·29-s + 4·31-s + 2·37-s − 4·39-s + 10·41-s − 43-s − 12·47-s − 7·49-s − 2·53-s + 4·57-s − 8·59-s + 12·61-s + 8·67-s − 8·69-s − 12·71-s + 6·73-s + 81-s + 6·83-s − 2·87-s + 12·89-s − 4·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.10·13-s − 0.917·19-s + 1.66·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.640·39-s + 1.56·41-s − 0.152·43-s − 1.75·47-s − 49-s − 0.274·53-s + 0.529·57-s − 1.04·59-s + 1.53·61-s + 0.977·67-s − 0.963·69-s − 1.42·71-s + 0.702·73-s + 1/9·81-s + 0.658·83-s − 0.214·87-s + 1.27·89-s − 0.414·93-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

\(L(1)\) \(\approx\) \(2.147312115\)
\(L(\frac12)\) \(\approx\) \(2.147312115\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.43654547608076, −14.05938741975442, −13.20086255393084, −12.94451166986815, −12.66563854102634, −11.71112900428035, −11.42051771284825, −10.92824297509402, −10.49981599662770, −9.896422931993983, −9.246940205063623, −8.788854736505035, −8.178347319833069, −7.697609811812724, −6.896678061562033, −6.386226640459531, −6.140136393846844, −5.273302173627833, −4.779708923747537, −4.235450341077553, −3.473502174950433, −2.903913784278444, −2.026304598589618, −1.221669473830884, −0.5927607931636464, 0.5927607931636464, 1.221669473830884, 2.026304598589618, 2.903913784278444, 3.473502174950433, 4.235450341077553, 4.779708923747537, 5.273302173627833, 6.140136393846844, 6.386226640459531, 6.896678061562033, 7.697609811812724, 8.178347319833069, 8.788854736505035, 9.246940205063623, 9.896422931993983, 10.49981599662770, 10.92824297509402, 11.42051771284825, 11.71112900428035, 12.66563854102634, 12.94451166986815, 13.20086255393084, 14.05938741975442, 14.43654547608076

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