Properties

Label 2-51600-1.1-c1-0-24
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 + 4·7-s + 9-s + 2·11-s − 2·13-s − 6·19-s − 4·21-s − 27-s + 10·29-s − 8·31-s − 2·33-s + 4·37-s + 2·39-s − 10·41-s − 43-s + 9·49-s − 12·53-s + 6·57-s + 6·59-s − 10·61-s + 4·63-s + 12·67-s − 4·71-s + 8·73-s + 8·77-s + 16·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.37·19-s − 0.872·21-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.348·33-s + 0.657·37-s + 0.320·39-s − 1.56·41-s − 0.152·43-s + 9/7·49-s − 1.64·53-s + 0.794·57-s + 0.781·59-s − 1.28·61-s + 0.503·63-s + 1.46·67-s − 0.474·71-s + 0.936·73-s + 0.911·77-s + 1.80·79-s + 1/9·81-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

\(L(1)\) \(\approx\) \(1.962935671\)
\(L(\frac12)\) \(\approx\) \(1.962935671\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.49468267765332, −14.08810030997948, −13.56638420433977, −12.73220055250599, −12.41150912697524, −11.91425313595242, −11.30088961171928, −11.02974727621944, −10.48943427711687, −9.920181674909167, −9.296721773549003, −8.609378731861663, −8.206244166175732, −7.741472359624609, −6.904114616203141, −6.616807181477894, −5.910543430917178, −5.190285266964913, −4.770500590606104, −4.334453162074023, −3.647941009017879, −2.695918432550934, −1.890860050634127, −1.483084623967288, −0.5124688515613306, 0.5124688515613306, 1.483084623967288, 1.890860050634127, 2.695918432550934, 3.647941009017879, 4.334453162074023, 4.770500590606104, 5.190285266964913, 5.910543430917178, 6.616807181477894, 6.904114616203141, 7.741472359624609, 8.206244166175732, 8.609378731861663, 9.296721773549003, 9.920181674909167, 10.48943427711687, 11.02974727621944, 11.30088961171928, 11.91425313595242, 12.41150912697524, 12.73220055250599, 13.56638420433977, 14.08810030997948, 14.49468267765332

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