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  + 28.7·3-s + 25·5-s + 42.1·7-s + 581.·9-s − 416.·11-s + 966.·13-s + 717.·15-s − 1.83e3·17-s − 317.·19-s + 1.21e3·21-s − 1.56e3·23-s + 625·25-s + 9.72e3·27-s + 7.75e3·29-s − 102.·31-s − 1.19e4·33-s + 1.05e3·35-s + 1.93e3·37-s + 2.77e4·39-s + 7.99e3·41-s − 1.65e4·43-s + 1.45e4·45-s − 1.86e4·47-s − 1.50e4·49-s − 5.26e4·51-s − 1.49e4·53-s − 1.04e4·55-s + ⋯
L(s)  = 1  + 1.84·3-s + 0.447·5-s + 0.325·7-s + 2.39·9-s − 1.03·11-s + 1.58·13-s + 0.823·15-s − 1.53·17-s − 0.201·19-s + 0.598·21-s − 0.618·23-s + 0.200·25-s + 2.56·27-s + 1.71·29-s − 0.0191·31-s − 1.91·33-s + 0.145·35-s + 0.232·37-s + 2.92·39-s + 0.742·41-s − 1.36·43-s + 1.07·45-s − 1.23·47-s − 0.894·49-s − 2.83·51-s − 0.732·53-s − 0.463·55-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

\(L(3)\) \(\approx\) \(3.54135\)
\(L(\frac12)\) \(\approx\) \(3.54135\)
\(L(\frac{7}{2})\) 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 \( 1 - 28.7T + 243T^{2} \)
7 \( 1 - 42.1T + 1.68e4T^{2} \)
11 \( 1 + 416.T + 1.61e5T^{2} \)
13 \( 1 - 966.T + 3.71e5T^{2} \)
17 \( 1 + 1.83e3T + 1.41e6T^{2} \)
19 \( 1 + 317.T + 2.47e6T^{2} \)
23 \( 1 + 1.56e3T + 6.43e6T^{2} \)
29 \( 1 - 7.75e3T + 2.05e7T^{2} \)
31 \( 1 + 102.T + 2.86e7T^{2} \)
37 \( 1 - 1.93e3T + 6.93e7T^{2} \)
41 \( 1 - 7.99e3T + 1.15e8T^{2} \)
43 \( 1 + 1.65e4T + 1.47e8T^{2} \)
47 \( 1 + 1.86e4T + 2.29e8T^{2} \)
53 \( 1 + 1.49e4T + 4.18e8T^{2} \)
59 \( 1 + 1.98e4T + 7.14e8T^{2} \)
61 \( 1 + 1.80e4T + 8.44e8T^{2} \)
67 \( 1 - 5.50e4T + 1.35e9T^{2} \)
71 \( 1 + 1.12e4T + 1.80e9T^{2} \)
73 \( 1 + 4.01e3T + 2.07e9T^{2} \)
79 \( 1 + 2.40e4T + 3.07e9T^{2} \)
83 \( 1 + 7.05e4T + 3.93e9T^{2} \)
89 \( 1 + 6.07e4T + 5.58e9T^{2} \)
97 \( 1 + 3.11e4T + 8.58e9T^{2} \)
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

−13.52264286000985000523930192144, −12.92523797441355372694644036642, −10.95955500367018888610578644902, −9.843976300791482150622432771857, −8.622647609257301174494213539937, −8.119619806704461866264543309587, −6.54488774356521669943509936710, −4.47386861212425605095010252040, −3.00116954620236432032596373564, −1.76865176153048056221471300631, 1.76865176153048056221471300631, 3.00116954620236432032596373564, 4.47386861212425605095010252040, 6.54488774356521669943509936710, 8.119619806704461866264543309587, 8.622647609257301174494213539937, 9.843976300791482150622432771857, 10.95955500367018888610578644902, 12.92523797441355372694644036642, 13.52264286000985000523930192144

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