Properties

Label 2-51600-1.1-c1-0-34
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 + 5·11-s + 3·13-s + 3·17-s − 4·19-s − 7·23-s + 27-s − 8·29-s + 3·31-s + 5·33-s + 3·39-s + 3·41-s + 43-s − 7·49-s + 3·51-s + 9·53-s − 4·57-s − 8·59-s − 11·67-s − 7·69-s − 8·71-s + 4·73-s + 8·79-s + 81-s + 9·83-s − 8·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.832·13-s + 0.727·17-s − 0.917·19-s − 1.45·23-s + 0.192·27-s − 1.48·29-s + 0.538·31-s + 0.870·33-s + 0.480·39-s + 0.468·41-s + 0.152·43-s − 49-s + 0.420·51-s + 1.23·53-s − 0.529·57-s − 1.04·59-s − 1.34·67-s − 0.842·69-s − 0.949·71-s + 0.468·73-s + 0.900·79-s + 1/9·81-s + 0.987·83-s − 0.857·87-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

\(L(1)\) \(\approx\) \(3.390163552\)
\(L(\frac12)\) \(\approx\) \(3.390163552\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−14.58845113922975, −13.91331075462239, −13.63757386580412, −13.01639472940643, −12.39207371194906, −11.98112320051437, −11.48042185126061, −10.83819193952157, −10.37110277581454, −9.609892614304986, −9.366262511721385, −8.711655338771452, −8.277407042558226, −7.710081071053691, −7.137139244458314, −6.378756610977421, −6.095992848780582, −5.469169427484887, −4.426013305603307, −4.071124147800235, −3.577916911646758, −2.922338899428035, −1.892229815154906, −1.606724467257259, −0.6234051484986086, 0.6234051484986086, 1.606724467257259, 1.892229815154906, 2.922338899428035, 3.577916911646758, 4.071124147800235, 4.426013305603307, 5.469169427484887, 6.095992848780582, 6.378756610977421, 7.137139244458314, 7.710081071053691, 8.277407042558226, 8.711655338771452, 9.366262511721385, 9.609892614304986, 10.37110277581454, 10.83819193952157, 11.48042185126061, 11.98112320051437, 12.39207371194906, 13.01639472940643, 13.63757386580412, 13.91331075462239, 14.58845113922975

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