Properties

Label 2-51600-1.1-c1-0-48
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 + 4·13-s + 6·17-s − 2·19-s − 21-s + 9·23-s + 27-s + 6·29-s − 2·31-s − 2·37-s + 4·39-s + 6·41-s + 43-s − 9·47-s − 6·49-s + 6·51-s + 6·53-s − 2·57-s + 12·59-s + 2·61-s − 63-s + 5·67-s + 9·69-s + 12·71-s + 4·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.458·19-s − 0.218·21-s + 1.87·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 0.640·39-s + 0.937·41-s + 0.152·43-s − 1.31·47-s − 6/7·49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s + 1.56·59-s + 0.256·61-s − 0.125·63-s + 0.610·67-s + 1.08·69-s + 1.42·71-s + 0.468·73-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

\(L(1)\) \(\approx\) \(4.088380757\)
\(L(\frac12)\) \(\approx\) \(4.088380757\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.44053016881402, −14.14079426730202, −13.31397161473711, −13.01811567400215, −12.66512132323849, −11.93075571976766, −11.40524719934442, −10.80704882513767, −10.35520546848880, −9.764117038018410, −9.283344465655595, −8.666867074717271, −8.317845667749208, −7.698665591014782, −7.091442556706274, −6.511219371519458, −6.068854930911047, −5.186395351231843, −4.878251048402594, −3.802107777392311, −3.545810307652295, −2.908327970348489, −2.219297838315977, −1.232277139778317, −0.7830632128657576, 0.7830632128657576, 1.232277139778317, 2.219297838315977, 2.908327970348489, 3.545810307652295, 3.802107777392311, 4.878251048402594, 5.186395351231843, 6.068854930911047, 6.511219371519458, 7.091442556706274, 7.698665591014782, 8.317845667749208, 8.666867074717271, 9.283344465655595, 9.764117038018410, 10.35520546848880, 10.80704882513767, 11.40524719934442, 11.93075571976766, 12.66512132323849, 13.01811567400215, 13.31397161473711, 14.14079426730202, 14.44053016881402

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