Properties

Degree $2$
Conductor $576$
Sign $1$
Motivic weight $7$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 82·5-s + 456·7-s + 2.52e3·11-s + 1.07e4·13-s + 1.11e4·17-s + 4.12e3·19-s + 8.17e4·23-s − 7.14e4·25-s + 9.97e4·29-s + 4.04e4·31-s − 3.73e4·35-s + 4.19e5·37-s − 1.41e5·41-s − 6.90e5·43-s − 6.82e5·47-s − 6.15e5·49-s + 1.81e6·53-s − 2.06e5·55-s + 9.66e5·59-s − 1.88e6·61-s − 8.83e5·65-s + 2.96e6·67-s − 2.54e6·71-s − 1.68e6·73-s + 1.15e6·77-s − 4.03e6·79-s + 5.38e6·83-s + ⋯
L(s)  = 1  − 0.293·5-s + 0.502·7-s + 0.571·11-s + 1.36·13-s + 0.550·17-s + 0.137·19-s + 1.40·23-s − 0.913·25-s + 0.759·29-s + 0.244·31-s − 0.147·35-s + 1.36·37-s − 0.320·41-s − 1.32·43-s − 0.958·47-s − 0.747·49-s + 1.67·53-s − 0.167·55-s + 0.612·59-s − 1.06·61-s − 0.399·65-s + 1.20·67-s − 0.844·71-s − 0.505·73-s + 0.287·77-s − 0.921·79-s + 1.03·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $1$
Motivic weight: \(7\)
Character: $\chi_{576} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.917943943\)
\(L(\frac12)\) \(\approx\) \(2.917943943\)
\(L(\frac{9}{2})\) 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 \( 1 \)
good5 \( 1 + 82 T + p^{7} T^{2} \)
7 \( 1 - 456 T + p^{7} T^{2} \)
11 \( 1 - 2524 T + p^{7} T^{2} \)
13 \( 1 - 10778 T + p^{7} T^{2} \)
17 \( 1 - 11150 T + p^{7} T^{2} \)
19 \( 1 - 4124 T + p^{7} T^{2} \)
23 \( 1 - 81704 T + p^{7} T^{2} \)
29 \( 1 - 99798 T + p^{7} T^{2} \)
31 \( 1 - 40480 T + p^{7} T^{2} \)
37 \( 1 - 419442 T + p^{7} T^{2} \)
41 \( 1 + 141402 T + p^{7} T^{2} \)
43 \( 1 + 690428 T + p^{7} T^{2} \)
47 \( 1 + 682032 T + p^{7} T^{2} \)
53 \( 1 - 1813118 T + p^{7} T^{2} \)
59 \( 1 - 966028 T + p^{7} T^{2} \)
61 \( 1 + 1887670 T + p^{7} T^{2} \)
67 \( 1 - 2965868 T + p^{7} T^{2} \)
71 \( 1 + 2548232 T + p^{7} T^{2} \)
73 \( 1 + 1680326 T + p^{7} T^{2} \)
79 \( 1 + 4038064 T + p^{7} T^{2} \)
83 \( 1 - 5385764 T + p^{7} T^{2} \)
89 \( 1 - 6473046 T + p^{7} T^{2} \)
97 \( 1 + 6065758 T + p^{7} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.552975603232897470848418589867, −8.598526845692192510880991757679, −7.980260997738736541829598682213, −6.88808253992869487355510280827, −6.02390550697264592778402047490, −4.96057308224437440249545068621, −3.95044708842449749935144025533, −3.05521636261578188440299484107, −1.58975437713899942455776316206, −0.794075146025589823544590102017, 0.794075146025589823544590102017, 1.58975437713899942455776316206, 3.05521636261578188440299484107, 3.95044708842449749935144025533, 4.96057308224437440249545068621, 6.02390550697264592778402047490, 6.88808253992869487355510280827, 7.980260997738736541829598682213, 8.598526845692192510880991757679, 9.552975603232897470848418589867

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