Properties

Label 2-51600-1.1-c1-0-2
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 − 6·11-s − 2·13-s + 2·19-s − 4·23-s − 27-s + 2·29-s − 8·31-s + 6·33-s + 8·37-s + 2·39-s − 2·41-s − 43-s + 12·47-s − 7·49-s − 8·53-s − 2·57-s − 2·59-s − 6·61-s − 4·67-s + 4·69-s − 12·71-s + 81-s + 12·83-s − 2·87-s + 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.458·19-s − 0.834·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 1.04·33-s + 1.31·37-s + 0.320·39-s − 0.312·41-s − 0.152·43-s + 1.75·47-s − 49-s − 1.09·53-s − 0.264·57-s − 0.260·59-s − 0.768·61-s − 0.488·67-s + 0.481·69-s − 1.42·71-s + 1/9·81-s + 1.31·83-s − 0.214·87-s + 0.635·89-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

\(L(1)\) \(\approx\) \(0.4368757324\)
\(L(\frac12)\) \(\approx\) \(0.4368757324\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.59314637823498, −13.83541381667747, −13.43242774590558, −12.91283131155943, −12.38343049699246, −12.08622457332091, −11.21870765417114, −11.00131298153330, −10.27510760798150, −10.04576122098121, −9.363411217965190, −8.792643748563307, −7.956423989426050, −7.604946384528247, −7.300167230021830, −6.313624116554137, −5.945554535359522, −5.255279413020455, −4.913660672466673, −4.261538508272923, −3.452019148454616, −2.730445778350844, −2.198943720863848, −1.317860449767635, −0.2408782833768973, 0.2408782833768973, 1.317860449767635, 2.198943720863848, 2.730445778350844, 3.452019148454616, 4.261538508272923, 4.913660672466673, 5.255279413020455, 5.945554535359522, 6.313624116554137, 7.300167230021830, 7.604946384528247, 7.956423989426050, 8.792643748563307, 9.363411217965190, 10.04576122098121, 10.27510760798150, 11.00131298153330, 11.21870765417114, 12.08622457332091, 12.38343049699246, 12.91283131155943, 13.43242774590558, 13.83541381667747, 14.59314637823498

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