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  − 22·3-s − 25·5-s − 218·7-s + 241·9-s + 480·11-s − 622·13-s + 550·15-s + 186·17-s + 1.20e3·19-s + 4.79e3·21-s + 3.18e3·23-s + 625·25-s + 44·27-s + 5.52e3·29-s − 9.35e3·31-s − 1.05e4·33-s + 5.45e3·35-s + 5.61e3·37-s + 1.36e4·39-s − 1.43e4·41-s + 370·43-s − 6.02e3·45-s − 1.61e4·47-s + 3.07e4·49-s − 4.09e3·51-s − 4.37e3·53-s − 1.20e4·55-s + ⋯
L(s)  = 1  − 1.41·3-s − 0.447·5-s − 1.68·7-s + 0.991·9-s + 1.19·11-s − 1.02·13-s + 0.631·15-s + 0.156·17-s + 0.765·19-s + 2.37·21-s + 1.25·23-s + 1/5·25-s + 0.0116·27-s + 1.22·29-s − 1.74·31-s − 1.68·33-s + 0.752·35-s + 0.674·37-s + 1.44·39-s − 1.33·41-s + 0.0305·43-s − 0.443·45-s − 1.06·47-s + 1.82·49-s − 0.220·51-s − 0.213·53-s − 0.534·55-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  =  \((2,\ 80,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(0.623036\)
\(L(\frac12)\)  \(\approx\)  \(0.623036\)
\(L(\frac{7}{2})\)   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 \( 1 + 22 T + p^{5} T^{2} \)
7 \( 1 + 218 T + p^{5} T^{2} \)
11 \( 1 - 480 T + p^{5} T^{2} \)
13 \( 1 + 622 T + p^{5} T^{2} \)
17 \( 1 - 186 T + p^{5} T^{2} \)
19 \( 1 - 1204 T + p^{5} T^{2} \)
23 \( 1 - 3186 T + p^{5} T^{2} \)
29 \( 1 - 5526 T + p^{5} T^{2} \)
31 \( 1 + 9356 T + p^{5} T^{2} \)
37 \( 1 - 5618 T + p^{5} T^{2} \)
41 \( 1 + 14394 T + p^{5} T^{2} \)
43 \( 1 - 370 T + p^{5} T^{2} \)
47 \( 1 + 16146 T + p^{5} T^{2} \)
53 \( 1 + 4374 T + p^{5} T^{2} \)
59 \( 1 - 11748 T + p^{5} T^{2} \)
61 \( 1 - 13202 T + p^{5} T^{2} \)
67 \( 1 - 11542 T + p^{5} T^{2} \)
71 \( 1 - 29532 T + p^{5} T^{2} \)
73 \( 1 - 33698 T + p^{5} T^{2} \)
79 \( 1 + 31208 T + p^{5} T^{2} \)
83 \( 1 - 38466 T + p^{5} T^{2} \)
89 \( 1 - 119514 T + p^{5} T^{2} \)
97 \( 1 - 94658 T + p^{5} T^{2} \)
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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

−12.99180423072000906025100238564, −12.20503587493974279058603668291, −11.43546142321154164328509956893, −10.14750686889858341477658672807, −9.210234300283595616961731312476, −7.09431922732523096374722704248, −6.39412849979514070350775092831, −5.04152291009129108696479856611, −3.39440251294930143411675588113, −0.62677463919458102093413935872, 0.62677463919458102093413935872, 3.39440251294930143411675588113, 5.04152291009129108696479856611, 6.39412849979514070350775092831, 7.09431922732523096374722704248, 9.210234300283595616961731312476, 10.14750686889858341477658672807, 11.43546142321154164328509956893, 12.20503587493974279058603668291, 12.99180423072000906025100238564

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