Properties

Label 2-51600-1.1-c1-0-13
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 − 2·7-s + 9-s + 6·13-s − 3·17-s − 2·19-s − 2·21-s − 3·23-s + 27-s − 4·29-s + 31-s − 10·37-s + 6·39-s + 3·41-s − 43-s − 7·47-s − 3·49-s − 3·51-s − 9·53-s − 2·57-s − 3·59-s − 2·61-s − 2·63-s + 11·67-s − 3·69-s − 14·71-s + 3·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.66·13-s − 0.727·17-s − 0.458·19-s − 0.436·21-s − 0.625·23-s + 0.192·27-s − 0.742·29-s + 0.179·31-s − 1.64·37-s + 0.960·39-s + 0.468·41-s − 0.152·43-s − 1.02·47-s − 3/7·49-s − 0.420·51-s − 1.23·53-s − 0.264·57-s − 0.390·59-s − 0.256·61-s − 0.251·63-s + 1.34·67-s − 0.361·69-s − 1.66·71-s + 0.337·79-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

\(L(1)\) \(\approx\) \(1.933546039\)
\(L(\frac12)\) \(\approx\) \(1.933546039\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.45321808331799, −13.85919477169545, −13.47194918693201, −13.04802280634032, −12.62992701412029, −11.96588218186651, −11.34441830753419, −10.79374361084678, −10.43909706200871, −9.711530581510352, −9.200692640599060, −8.835242488919892, −8.148468477808775, −7.889501076552205, −6.867988284617111, −6.606379151713716, −6.050192395724108, −5.427981269491059, −4.566682908108977, −4.003463721891949, −3.403200859788131, −3.035457250080344, −1.981200672015323, −1.598725618926986, −0.4531888134947494, 0.4531888134947494, 1.598725618926986, 1.981200672015323, 3.035457250080344, 3.403200859788131, 4.003463721891949, 4.566682908108977, 5.427981269491059, 6.050192395724108, 6.606379151713716, 6.867988284617111, 7.889501076552205, 8.148468477808775, 8.835242488919892, 9.200692640599060, 9.711530581510352, 10.43909706200871, 10.79374361084678, 11.34441830753419, 11.96588218186651, 12.62992701412029, 13.04802280634032, 13.47194918693201, 13.85919477169545, 14.45321808331799

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