Properties

Label 2-62400-1.1-c1-0-53
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 − 4·7-s + 9-s + 13-s − 2·17-s − 8·19-s + 4·21-s − 4·23-s − 27-s − 6·29-s − 8·31-s + 6·37-s − 39-s − 6·41-s − 4·43-s − 4·47-s + 9·49-s + 2·51-s + 6·53-s + 8·57-s − 8·59-s + 10·61-s − 4·63-s + 4·67-s + 4·69-s + 8·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 1.05·57-s − 1.04·59-s + 1.28·61-s − 0.503·63-s + 0.488·67-s + 0.481·69-s + 0.949·71-s + 1.63·73-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.61641930572398, −13.92056788113208, −13.22706746098219, −13.00684377836910, −12.63074208217091, −12.11827139652737, −11.31295355697834, −11.09511208323875, −10.35712131102272, −10.06485878933135, −9.378452620551565, −9.047515833833929, −8.342448652515348, −7.784104998090983, −6.978754661610830, −6.597988490771605, −6.209933537964895, −5.677479032978452, −5.029666583101827, −4.217444737723816, −3.756618222326893, −3.283942137840444, −2.209605684600834, −1.919148264993139, −0.6021625994337538, 0, 0.6021625994337538, 1.919148264993139, 2.209605684600834, 3.283942137840444, 3.756618222326893, 4.217444737723816, 5.029666583101827, 5.677479032978452, 6.209933537964895, 6.597988490771605, 6.978754661610830, 7.784104998090983, 8.342448652515348, 9.047515833833929, 9.378452620551565, 10.06485878933135, 10.35712131102272, 11.09511208323875, 11.31295355697834, 12.11827139652737, 12.63074208217091, 13.00684377836910, 13.22706746098219, 13.92056788113208, 14.61641930572398

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