Properties

Label 2-62400-1.1-c1-0-182
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s + 4·11-s − 13-s − 8·17-s − 6·19-s + 2·21-s + 6·23-s + 27-s + 4·29-s + 4·33-s − 2·37-s − 39-s − 2·41-s + 4·43-s − 3·49-s − 8·51-s − 10·53-s − 6·57-s + 4·59-s + 10·61-s + 2·63-s − 12·67-s + 6·69-s + 8·71-s + 8·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.94·17-s − 1.37·19-s + 0.436·21-s + 1.25·23-s + 0.192·27-s + 0.742·29-s + 0.696·33-s − 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 3/7·49-s − 1.12·51-s − 1.37·53-s − 0.794·57-s + 0.520·59-s + 1.28·61-s + 0.251·63-s − 1.46·67-s + 0.722·69-s + 0.949·71-s + 0.936·73-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: \(1\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.45936768846967, −14.22731550044571, −13.50740343900434, −12.99474220454888, −12.69552943222671, −11.90954673979685, −11.48050602363875, −10.89880026934887, −10.66100524789105, −9.820220766064453, −9.185972985549463, −8.922697619912190, −8.360694977376551, −8.022895677981988, −6.979911033406157, −6.840636842040231, −6.342620700373621, −5.471264962961606, −4.624233397766230, −4.463882144199261, −3.856086483714176, −3.011987318142309, −2.345949590442893, −1.797389169948226, −1.115452968796426, 0, 1.115452968796426, 1.797389169948226, 2.345949590442893, 3.011987318142309, 3.856086483714176, 4.463882144199261, 4.624233397766230, 5.471264962961606, 6.342620700373621, 6.840636842040231, 6.979911033406157, 8.022895677981988, 8.360694977376551, 8.922697619912190, 9.185972985549463, 9.820220766064453, 10.66100524789105, 10.89880026934887, 11.48050602363875, 11.90954673979685, 12.69552943222671, 12.99474220454888, 13.50740343900434, 14.22731550044571, 14.45936768846967

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