Properties

Label 2-62400-1.1-c1-0-51
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 + 7-s + 9-s − 5·11-s − 13-s + 7·17-s + 6·19-s + 21-s − 3·23-s + 27-s − 2·29-s + 2·31-s − 5·33-s + 7·37-s − 39-s + 9·41-s − 8·43-s − 10·47-s − 6·49-s + 7·51-s + 5·53-s + 6·57-s − 5·61-s + 63-s − 4·67-s − 3·69-s + 9·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 1.69·17-s + 1.37·19-s + 0.218·21-s − 0.625·23-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.870·33-s + 1.15·37-s − 0.160·39-s + 1.40·41-s − 1.21·43-s − 1.45·47-s − 6/7·49-s + 0.980·51-s + 0.686·53-s + 0.794·57-s − 0.640·61-s + 0.125·63-s − 0.488·67-s − 0.361·69-s + 1.06·71-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

\(L(1)\) \(\approx\) \(3.064354212\)
\(L(\frac12)\) \(\approx\) \(3.064354212\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−14.25964599785550, −13.83954612258243, −13.23248625949275, −12.90047627822448, −12.25669317781652, −11.78323948492841, −11.28382270580029, −10.58485819759998, −10.12129831581473, −9.622766681898388, −9.375539885631353, −8.313161629854991, −8.007910901048415, −7.689087952449269, −7.245998109174863, −6.353150925490609, −5.697377906696572, −5.197396857027037, −4.795777586497903, −3.959005173124724, −3.201314341632104, −2.912443787606390, −2.134951746291032, −1.386828418926796, −0.5856872351892453, 0.5856872351892453, 1.386828418926796, 2.134951746291032, 2.912443787606390, 3.201314341632104, 3.959005173124724, 4.795777586497903, 5.197396857027037, 5.697377906696572, 6.353150925490609, 7.245998109174863, 7.689087952449269, 8.007910901048415, 8.313161629854991, 9.375539885631353, 9.622766681898388, 10.12129831581473, 10.58485819759998, 11.28382270580029, 11.78323948492841, 12.25669317781652, 12.90047627822448, 13.23248625949275, 13.83954612258243, 14.25964599785550

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