Properties

Label 2-51600-1.1-c1-0-14
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 − 4·11-s + 2·13-s − 6·17-s + 8·19-s − 4·21-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 2·37-s + 2·39-s + 2·41-s − 43-s + 8·47-s + 9·49-s − 6·51-s − 6·53-s + 8·57-s − 12·59-s + 10·61-s − 4·63-s + 4·67-s + 2·73-s + 16·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 0.872·21-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.152·43-s + 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 1.28·61-s − 0.503·63-s + 0.488·67-s + 0.234·73-s + 1.82·77-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

\(L(1)\) \(\approx\) \(1.745168056\)
\(L(\frac12)\) \(\approx\) \(1.745168056\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.31814863267625, −13.78954172434811, −13.44102999143365, −13.16474863114967, −12.52775260196485, −12.09361412189338, −11.34235428384857, −10.81598083526215, −10.24876122646310, −9.741177729578175, −9.358643682628033, −8.835915182441885, −8.136728885822075, −7.750004422453851, −6.976436562886364, −6.634412273350660, −5.983816533916641, −5.386413631050907, −4.677075208086431, −4.016259510336541, −3.339036279071241, −2.738407331827265, −2.512701807798371, −1.316543746588320, −0.4577705328317476, 0.4577705328317476, 1.316543746588320, 2.512701807798371, 2.738407331827265, 3.339036279071241, 4.016259510336541, 4.677075208086431, 5.386413631050907, 5.983816533916641, 6.634412273350660, 6.976436562886364, 7.750004422453851, 8.136728885822075, 8.835915182441885, 9.358643682628033, 9.741177729578175, 10.24876122646310, 10.81598083526215, 11.34235428384857, 12.09361412189338, 12.52775260196485, 13.16474863114967, 13.44102999143365, 13.78954172434811, 14.31814863267625

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