Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 25·5-s + 118·7-s − 207·9-s − 192·11-s + 1.10e3·13-s + 150·15-s + 762·17-s + 2.74e3·19-s − 708·21-s − 1.56e3·23-s + 625·25-s + 2.70e3·27-s + 5.91e3·29-s + 6.86e3·31-s + 1.15e3·33-s − 2.95e3·35-s − 5.51e3·37-s − 6.63e3·39-s − 378·41-s + 2.43e3·43-s + 5.17e3·45-s − 1.31e4·47-s − 2.88e3·49-s − 4.57e3·51-s − 9.17e3·53-s + 4.80e3·55-s + ⋯
L(s)  = 1  − 0.384·3-s − 0.447·5-s + 0.910·7-s − 0.851·9-s − 0.478·11-s + 1.81·13-s + 0.172·15-s + 0.639·17-s + 1.74·19-s − 0.350·21-s − 0.617·23-s + 1/5·25-s + 0.712·27-s + 1.30·29-s + 1.28·31-s + 0.184·33-s − 0.407·35-s − 0.662·37-s − 0.698·39-s − 0.0351·41-s + 0.200·43-s + 0.380·45-s − 0.866·47-s − 0.171·49-s − 0.246·51-s − 0.448·53-s + 0.213·55-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

\(L(3)\) \(\approx\) \(1.57986\)
\(L(\frac12)\) \(\approx\) \(1.57986\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{2} T \)
good3 \( 1 + 2 p T + p^{5} T^{2} \)
7 \( 1 - 118 T + p^{5} T^{2} \)
11 \( 1 + 192 T + p^{5} T^{2} \)
13 \( 1 - 1106 T + p^{5} T^{2} \)
17 \( 1 - 762 T + p^{5} T^{2} \)
19 \( 1 - 2740 T + p^{5} T^{2} \)
23 \( 1 + 1566 T + p^{5} T^{2} \)
29 \( 1 - 5910 T + p^{5} T^{2} \)
31 \( 1 - 6868 T + p^{5} T^{2} \)
37 \( 1 + 5518 T + p^{5} T^{2} \)
41 \( 1 + 378 T + p^{5} T^{2} \)
43 \( 1 - 2434 T + p^{5} T^{2} \)
47 \( 1 + 13122 T + p^{5} T^{2} \)
53 \( 1 + 9174 T + p^{5} T^{2} \)
59 \( 1 - 34980 T + p^{5} T^{2} \)
61 \( 1 + 9838 T + p^{5} T^{2} \)
67 \( 1 + 33722 T + p^{5} T^{2} \)
71 \( 1 + 70212 T + p^{5} T^{2} \)
73 \( 1 - 21986 T + p^{5} T^{2} \)
79 \( 1 + 4520 T + p^{5} T^{2} \)
83 \( 1 - 109074 T + p^{5} T^{2} \)
89 \( 1 - 38490 T + p^{5} T^{2} \)
97 \( 1 + 1918 T + p^{5} T^{2} \)
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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

−13.59637353593785418494428048946, −11.95171120194205814771989789563, −11.41016682502785427631918916779, −10.32532415177369680032614476158, −8.608934802210214958856773811804, −7.84895588795961532318692449177, −6.13315518073258561277170285807, −4.97495882515339558163137817856, −3.26690440073859703128925123776, −1.04238978570574419378469037838, 1.04238978570574419378469037838, 3.26690440073859703128925123776, 4.97495882515339558163137817856, 6.13315518073258561277170285807, 7.84895588795961532318692449177, 8.608934802210214958856773811804, 10.32532415177369680032614476158, 11.41016682502785427631918916779, 11.95171120194205814771989789563, 13.59637353593785418494428048946

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