Properties

Degree $2$
Conductor $80$
Sign $1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 16.7·3-s + 25·5-s − 94.1·7-s + 36.4·9-s − 143.·11-s + 421.·13-s − 417.·15-s + 1.98e3·17-s + 1.31e3·19-s + 1.57e3·21-s + 4.02e3·23-s + 625·25-s + 3.45e3·27-s − 6.41e3·29-s + 2.35e3·31-s + 2.40e3·33-s − 2.35e3·35-s − 7.87e3·37-s − 7.04e3·39-s + 1.50e4·41-s − 1.14e3·43-s + 910.·45-s + 2.15e4·47-s − 7.94e3·49-s − 3.31e4·51-s + 9.56e3·53-s − 3.59e3·55-s + ⋯
L(s)  = 1  − 1.07·3-s + 0.447·5-s − 0.726·7-s + 0.149·9-s − 0.358·11-s + 0.691·13-s − 0.479·15-s + 1.66·17-s + 0.837·19-s + 0.778·21-s + 1.58·23-s + 0.200·25-s + 0.911·27-s − 1.41·29-s + 0.439·31-s + 0.383·33-s − 0.324·35-s − 0.945·37-s − 0.741·39-s + 1.40·41-s − 0.0941·43-s + 0.0670·45-s + 1.42·47-s − 0.472·49-s − 1.78·51-s + 0.467·53-s − 0.160·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $1$
Motivic weight: \(5\)
Character: $\chi_{80} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.18777\)
\(L(\frac12)\) \(\approx\) \(1.18777\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 25T \)
good3 \( 1 + 16.7T + 243T^{2} \)
7 \( 1 + 94.1T + 1.68e4T^{2} \)
11 \( 1 + 143.T + 1.61e5T^{2} \)
13 \( 1 - 421.T + 3.71e5T^{2} \)
17 \( 1 - 1.98e3T + 1.41e6T^{2} \)
19 \( 1 - 1.31e3T + 2.47e6T^{2} \)
23 \( 1 - 4.02e3T + 6.43e6T^{2} \)
29 \( 1 + 6.41e3T + 2.05e7T^{2} \)
31 \( 1 - 2.35e3T + 2.86e7T^{2} \)
37 \( 1 + 7.87e3T + 6.93e7T^{2} \)
41 \( 1 - 1.50e4T + 1.15e8T^{2} \)
43 \( 1 + 1.14e3T + 1.47e8T^{2} \)
47 \( 1 - 2.15e4T + 2.29e8T^{2} \)
53 \( 1 - 9.56e3T + 4.18e8T^{2} \)
59 \( 1 + 4.27e4T + 7.14e8T^{2} \)
61 \( 1 - 3.21e4T + 8.44e8T^{2} \)
67 \( 1 - 3.03e4T + 1.35e9T^{2} \)
71 \( 1 + 3.60e4T + 1.80e9T^{2} \)
73 \( 1 + 6.34e4T + 2.07e9T^{2} \)
79 \( 1 - 8.99e4T + 3.07e9T^{2} \)
83 \( 1 + 3.82e4T + 3.93e9T^{2} \)
89 \( 1 - 5.74e3T + 5.58e9T^{2} \)
97 \( 1 - 1.78e5T + 8.58e9T^{2} \)
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

−13.22141430387596731804573087869, −12.30307350352262584450903273939, −11.19926942396042857658251164258, −10.24098021989322024302160726647, −9.100889102275110912313548705786, −7.38097935374464368211514605799, −6.03448109439684576404105985846, −5.27784775488046631804627170362, −3.22327427245931833527234206242, −0.894891540791953808883245827389, 0.894891540791953808883245827389, 3.22327427245931833527234206242, 5.27784775488046631804627170362, 6.03448109439684576404105985846, 7.38097935374464368211514605799, 9.100889102275110912313548705786, 10.24098021989322024302160726647, 11.19926942396042857658251164258, 12.30307350352262584450903273939, 13.22141430387596731804573087869

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