Properties

Label 2-51600-1.1-c1-0-5
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 − 3·7-s + 9-s − 4·11-s + 3·13-s − 7·19-s − 3·21-s − 4·23-s + 27-s + 29-s − 3·31-s − 4·33-s − 12·37-s + 3·39-s + 9·41-s + 43-s + 6·47-s + 2·49-s + 6·53-s − 7·57-s + 4·59-s + 3·61-s − 3·63-s + 67-s − 4·69-s + 10·71-s − 11·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.20·11-s + 0.832·13-s − 1.60·19-s − 0.654·21-s − 0.834·23-s + 0.192·27-s + 0.185·29-s − 0.538·31-s − 0.696·33-s − 1.97·37-s + 0.480·39-s + 1.40·41-s + 0.152·43-s + 0.875·47-s + 2/7·49-s + 0.824·53-s − 0.927·57-s + 0.520·59-s + 0.384·61-s − 0.377·63-s + 0.122·67-s − 0.481·69-s + 1.18·71-s − 1.28·73-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

\(L(1)\) \(\approx\) \(0.9287664356\)
\(L(\frac12)\) \(\approx\) \(0.9287664356\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−14.35949341427739, −13.96928991309226, −13.37544480616134, −13.00667995159530, −12.55669588583417, −12.21719572770558, −11.23592513490379, −10.78780264210695, −10.25984617861658, −9.957461621352588, −9.215990364449782, −8.642228970480609, −8.371698315682912, −7.693515830757951, −6.998283778810100, −6.648849522654739, −5.778153458778691, −5.621569686701926, −4.574372682386526, −3.951394123638506, −3.554040479864836, −2.695175888156611, −2.350709404555666, −1.478155682006431, −0.3112254970830028, 0.3112254970830028, 1.478155682006431, 2.350709404555666, 2.695175888156611, 3.554040479864836, 3.951394123638506, 4.574372682386526, 5.621569686701926, 5.778153458778691, 6.648849522654739, 6.998283778810100, 7.693515830757951, 8.371698315682912, 8.642228970480609, 9.215990364449782, 9.957461621352588, 10.25984617861658, 10.78780264210695, 11.23592513490379, 12.21719572770558, 12.55669588583417, 13.00667995159530, 13.37544480616134, 13.96928991309226, 14.35949341427739

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