Properties

Label 2-62400-1.1-c1-0-19
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 − 4·7-s + 9-s − 13-s + 3·19-s + 4·21-s + 4·23-s − 27-s + 29-s − 8·31-s − 37-s + 39-s + 41-s + 6·43-s + 11·47-s + 9·49-s − 3·53-s − 3·57-s − 10·59-s − 4·61-s − 4·63-s + 13·67-s − 4·69-s + 9·71-s − 3·79-s + 81-s + 2·83-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.688·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 0.185·29-s − 1.43·31-s − 0.164·37-s + 0.160·39-s + 0.156·41-s + 0.914·43-s + 1.60·47-s + 9/7·49-s − 0.412·53-s − 0.397·57-s − 1.30·59-s − 0.512·61-s − 0.503·63-s + 1.58·67-s − 0.481·69-s + 1.06·71-s − 0.337·79-s + 1/9·81-s + 0.219·83-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

\(L(1)\) \(\approx\) \(1.114535169\)
\(L(\frac12)\) \(\approx\) \(1.114535169\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.15469020632573, −13.77196053213598, −12.98422070352300, −12.86071519323135, −12.26212881210631, −11.91228890452206, −11.06375994657626, −10.81377393945164, −10.18531332224687, −9.662067501990419, −9.138960915196158, −8.949425522230653, −7.834939321545447, −7.439020817381250, −6.879138659520360, −6.417189617074209, −5.836452109680877, −5.349373877765472, −4.751973807422505, −3.926096519099231, −3.464299538560782, −2.831106113187792, −2.145187380053560, −1.114143992479107, −0.4176011193059757, 0.4176011193059757, 1.114143992479107, 2.145187380053560, 2.831106113187792, 3.464299538560782, 3.926096519099231, 4.751973807422505, 5.349373877765472, 5.836452109680877, 6.417189617074209, 6.879138659520360, 7.439020817381250, 7.834939321545447, 8.949425522230653, 9.138960915196158, 9.662067501990419, 10.18531332224687, 10.81377393945164, 11.06375994657626, 11.91228890452206, 12.26212881210631, 12.86071519323135, 12.98422070352300, 13.77196053213598, 14.15469020632573

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