Properties

Label 2-51600-1.1-c1-0-31
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 + 9-s − 4·11-s + 6·17-s + 2·19-s − 6·23-s + 27-s + 4·29-s + 4·31-s − 4·33-s + 6·37-s + 10·41-s + 43-s − 6·47-s − 7·49-s + 6·51-s − 6·53-s + 2·57-s + 4·59-s + 10·61-s + 8·67-s − 6·69-s − 12·71-s + 2·73-s − 4·79-s + 81-s + 2·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.45·17-s + 0.458·19-s − 1.25·23-s + 0.192·27-s + 0.742·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s + 1.56·41-s + 0.152·43-s − 0.875·47-s − 49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s + 0.520·59-s + 1.28·61-s + 0.977·67-s − 0.722·69-s − 1.42·71-s + 0.234·73-s − 0.450·79-s + 1/9·81-s + 0.219·83-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

\(L(1)\) \(\approx\) \(2.906128691\)
\(L(\frac12)\) \(\approx\) \(2.906128691\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−14.58498766415481, −14.06575733014520, −13.43531872118418, −13.03469118659411, −12.48283843041112, −12.03101145736416, −11.42287914820513, −10.83025157818980, −10.18444350862168, −9.774313789870699, −9.547803532098916, −8.517432631774819, −8.164042030651564, −7.729758515639391, −7.341810369646173, −6.429674179000614, −5.942423270012033, −5.315017992816246, −4.743387936817684, −4.076140451897790, −3.365625687795635, −2.807063422090098, −2.290944403486044, −1.374013965554726, −0.5942855207298792, 0.5942855207298792, 1.374013965554726, 2.290944403486044, 2.807063422090098, 3.365625687795635, 4.076140451897790, 4.743387936817684, 5.315017992816246, 5.942423270012033, 6.429674179000614, 7.341810369646173, 7.729758515639391, 8.164042030651564, 8.517432631774819, 9.547803532098916, 9.774313789870699, 10.18444350862168, 10.83025157818980, 11.42287914820513, 12.03101145736416, 12.48283843041112, 13.03469118659411, 13.43531872118418, 14.06575733014520, 14.58498766415481

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