Properties

Degree 2
Conductor $ 2^{4} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s + 25·5-s + 108·7-s − 179·9-s + 604·11-s − 306·13-s + 200·15-s + 930·17-s + 1.32e3·19-s + 864·21-s + 852·23-s + 625·25-s − 3.37e3·27-s + 5.90e3·29-s + 3.32e3·31-s + 4.83e3·33-s + 2.70e3·35-s + 1.07e4·37-s − 2.44e3·39-s − 1.79e4·41-s − 9.26e3·43-s − 4.47e3·45-s + 9.79e3·47-s − 5.14e3·49-s + 7.44e3·51-s − 3.14e4·53-s + 1.51e4·55-s + ⋯
L(s)  = 1  + 0.513·3-s + 0.447·5-s + 0.833·7-s − 0.736·9-s + 1.50·11-s − 0.502·13-s + 0.229·15-s + 0.780·17-s + 0.841·19-s + 0.427·21-s + 0.335·23-s + 1/5·25-s − 0.891·27-s + 1.30·29-s + 0.620·31-s + 0.772·33-s + 0.372·35-s + 1.29·37-s − 0.257·39-s − 1.66·41-s − 0.764·43-s − 0.329·45-s + 0.646·47-s − 0.306·49-s + 0.400·51-s − 1.53·53-s + 0.673·55-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

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{80} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 80,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.51558\)
\(L(\frac12)\)  \(\approx\)  \(2.51558\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - p^{2} T \)
good3 \( 1 - 8 T + p^{5} T^{2} \)
7 \( 1 - 108 T + p^{5} T^{2} \)
11 \( 1 - 604 T + p^{5} T^{2} \)
13 \( 1 + 306 T + p^{5} T^{2} \)
17 \( 1 - 930 T + p^{5} T^{2} \)
19 \( 1 - 1324 T + p^{5} T^{2} \)
23 \( 1 - 852 T + p^{5} T^{2} \)
29 \( 1 - 5902 T + p^{5} T^{2} \)
31 \( 1 - 3320 T + p^{5} T^{2} \)
37 \( 1 - 10774 T + p^{5} T^{2} \)
41 \( 1 + 438 p T + p^{5} T^{2} \)
43 \( 1 + 9264 T + p^{5} T^{2} \)
47 \( 1 - 9796 T + p^{5} T^{2} \)
53 \( 1 + 31434 T + p^{5} T^{2} \)
59 \( 1 + 33228 T + p^{5} T^{2} \)
61 \( 1 + 40210 T + p^{5} T^{2} \)
67 \( 1 + 58864 T + p^{5} T^{2} \)
71 \( 1 - 55312 T + p^{5} T^{2} \)
73 \( 1 - 27258 T + p^{5} T^{2} \)
79 \( 1 + 31456 T + p^{5} T^{2} \)
83 \( 1 + 24552 T + p^{5} T^{2} \)
89 \( 1 + 90854 T + p^{5} T^{2} \)
97 \( 1 - 154706 T + p^{5} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.85273694860117436009073202056, −12.15958666634151428359040282183, −11.39429028737124444680867151070, −9.879716413386185058424037914854, −8.892103048762134208422541344453, −7.80409834275256631809765770801, −6.29198696662561509787258872707, −4.83541129581095762238082164593, −3.09561252845604081091670968037, −1.39329979838160468519123655450, 1.39329979838160468519123655450, 3.09561252845604081091670968037, 4.83541129581095762238082164593, 6.29198696662561509787258872707, 7.80409834275256631809765770801, 8.892103048762134208422541344453, 9.879716413386185058424037914854, 11.39429028737124444680867151070, 12.15958666634151428359040282183, 13.85273694860117436009073202056

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