Properties

Label 2-62400-1.1-c1-0-2
Degree $2$
Conductor $62400$
Sign $1$
Analytic cond. $498.266$
Root an. cond. $22.3218$
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 − 3·11-s − 13-s + 17-s − 6·19-s + 3·21-s − 5·23-s − 27-s + 6·29-s − 2·31-s + 3·33-s + 7·37-s + 39-s + 3·41-s + 8·43-s − 2·47-s + 2·49-s − 51-s − 53-s + 6·57-s + 15·61-s − 3·63-s − 12·67-s + 5·69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.242·17-s − 1.37·19-s + 0.654·21-s − 1.04·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.522·33-s + 1.15·37-s + 0.160·39-s + 0.468·41-s + 1.21·43-s − 0.291·47-s + 2/7·49-s − 0.140·51-s − 0.137·53-s + 0.794·57-s + 1.92·61-s − 0.377·63-s − 1.46·67-s + 0.601·69-s − 0.593·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(498.266\)
Root analytic conductor: \(22.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{62400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4559115548\)
\(L(\frac12)\) \(\approx\) \(0.4559115548\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−14.28702825132257, −13.62021615997577, −13.14957467291082, −12.66336960237870, −12.42095477464933, −11.83873730418067, −11.12381790735294, −10.68647534867369, −10.20164372513318, −9.780800066824217, −9.308642795956384, −8.539274384657253, −8.032886896491300, −7.508902664910926, −6.787451669564508, −6.389094555570501, −5.859153844876859, −5.406956608600845, −4.555474817432551, −4.168041254537628, −3.446741291109929, −2.595690186607982, −2.310451061857107, −1.167200693173935, −0.2526017776835508, 0.2526017776835508, 1.167200693173935, 2.310451061857107, 2.595690186607982, 3.446741291109929, 4.168041254537628, 4.555474817432551, 5.406956608600845, 5.859153844876859, 6.389094555570501, 6.787451669564508, 7.508902664910926, 8.032886896491300, 8.539274384657253, 9.308642795956384, 9.780800066824217, 10.20164372513318, 10.68647534867369, 11.12381790735294, 11.83873730418067, 12.42095477464933, 12.66336960237870, 13.14957467291082, 13.62021615997577, 14.28702825132257

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