Properties

Label 2-51600-1.1-c1-0-4
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 − 4·11-s − 4·13-s − 6·17-s − 6·19-s + 21-s − 23-s + 27-s + 6·29-s − 2·31-s − 4·33-s + 2·37-s − 4·39-s + 2·41-s + 43-s + 47-s − 6·49-s − 6·51-s − 6·53-s − 6·57-s − 8·59-s − 10·61-s + 63-s − 7·67-s − 69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 1.37·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.328·37-s − 0.640·39-s + 0.312·41-s + 0.152·43-s + 0.145·47-s − 6/7·49-s − 0.840·51-s − 0.824·53-s − 0.794·57-s − 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.855·67-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

\(L(1)\) \(\approx\) \(0.9634627714\)
\(L(\frac12)\) \(\approx\) \(0.9634627714\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−14.51877709104223, −14.04669095921363, −13.33035189081478, −13.11934042400439, −12.43095191587528, −12.16074456765963, −11.21829741341916, −10.86287634582067, −10.38348708626257, −9.862472671687409, −9.150353597592079, −8.806152701415371, −8.005270608551197, −7.891201703811737, −7.153472176008569, −6.536726348413837, −6.044658190704569, −5.135240583767356, −4.564684382548106, −4.411766933576322, −3.362827509366773, −2.607988030487885, −2.300167586594270, −1.605345439043598, −0.3061212897846791, 0.3061212897846791, 1.605345439043598, 2.300167586594270, 2.607988030487885, 3.362827509366773, 4.411766933576322, 4.564684382548106, 5.135240583767356, 6.044658190704569, 6.536726348413837, 7.153472176008569, 7.891201703811737, 8.005270608551197, 8.806152701415371, 9.150353597592079, 9.862472671687409, 10.38348708626257, 10.86287634582067, 11.21829741341916, 12.16074456765963, 12.43095191587528, 13.11934042400439, 13.33035189081478, 14.04669095921363, 14.51877709104223

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