Properties

Label 2-62400-1.1-c1-0-20
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 − 5·7-s + 9-s − 5·11-s − 13-s − 3·17-s + 4·19-s + 5·21-s + 5·23-s − 27-s + 4·29-s + 5·33-s + 7·37-s + 39-s + 11·41-s + 12·43-s + 6·47-s + 18·49-s + 3·51-s − 53-s − 4·57-s − 12·59-s + 7·61-s − 5·63-s − 4·67-s − 5·69-s − 7·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s + 1.09·21-s + 1.04·23-s − 0.192·27-s + 0.742·29-s + 0.870·33-s + 1.15·37-s + 0.160·39-s + 1.71·41-s + 1.82·43-s + 0.875·47-s + 18/7·49-s + 0.420·51-s − 0.137·53-s − 0.529·57-s − 1.56·59-s + 0.896·61-s − 0.629·63-s − 0.488·67-s − 0.601·69-s − 0.830·71-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

\(L(1)\) \(\approx\) \(0.9881681906\)
\(L(\frac12)\) \(\approx\) \(0.9881681906\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−14.06841834251549, −13.67836805489337, −12.99784026868604, −12.82127258692678, −12.49945571520379, −11.76233421783435, −11.15285543547952, −10.63269560403113, −10.26424860829858, −9.713136150650387, −9.191339335277397, −8.862250555530726, −7.837692323645034, −7.363994519322160, −7.060588858224556, −6.228312497618771, −5.892077171048054, −5.435595033819698, −4.567714737985983, −4.221956937238725, −3.172865260553151, −2.823038671928265, −2.365429694178106, −0.9844871708355888, −0.4228559436044133, 0.4228559436044133, 0.9844871708355888, 2.365429694178106, 2.823038671928265, 3.172865260553151, 4.221956937238725, 4.567714737985983, 5.435595033819698, 5.892077171048054, 6.228312497618771, 7.060588858224556, 7.363994519322160, 7.837692323645034, 8.862250555530726, 9.191339335277397, 9.713136150650387, 10.26424860829858, 10.63269560403113, 11.15285543547952, 11.76233421783435, 12.49945571520379, 12.82127258692678, 12.99784026868604, 13.67836805489337, 14.06841834251549

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