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Properties

Label 80-4000e40-1.1-c0e40-0-0
Degree $80$
Conductor $1.209\times 10^{144}$
Sign $1$
Analytic cond. $1.02019\times 10^{12}$
Root an. cond. $1.41289$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·89-s + 10·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  + 10·89-s + 10·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{200} \cdot 5^{120}\right)^{s/2} \, \Gamma_{\C}(s)^{40} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{200} \cdot 5^{120}\right)^{s/2} \, \Gamma_{\C}(s)^{40} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(80\)
Conductor: \(2^{200} \cdot 5^{120}\)
Sign: $1$
Analytic conductor: \(1.02019\times 10^{12}\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((80,\ 2^{200} \cdot 5^{120} ,\ ( \ : [0]^{40} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.479372882\)
\(L(\frac12)\) \(\approx\) \(2.479372882\)
\(L(1)\) 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 \( 1 - T^{10} + T^{20} - T^{30} + T^{40} \)
good3 \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
7 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5} \)
11 \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
13 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
17 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
19 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
23 \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
29 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
31 \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
41 \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
43 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{5} \)
47 \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
59 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
61 \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
67 \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
71 \( ( 1 - T^{10} + T^{20} - T^{30} + T^{40} )^{2} \)
73 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
79 \( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
83 \( 1 - T^{20} + T^{40} - T^{60} + T^{80} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{10}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \)
97 \( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2}( 1 - T^{10} + T^{20} - T^{30} + T^{40} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{80} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.26369159960921172172241333135, −1.25805989345304742219997418230, −1.21377494398929273802401269561, −1.15005556925190022366101250321, −1.14943795350724557058164068012, −1.14385158479943065221687485645, −1.02102102930052395706366641683, −0.999606620217287016226190991988, −0.936401792492719491646624141885, −0.925191814483574197454031239971, −0.923927910640083246920424549825, −0.915823152696376132682575643525, −0.915251977458336363664003567100, −0.846091326562409724932860429908, −0.818601724168504721975599420571, −0.76765634995372156350424394191, −0.61121635984475864695812019005, −0.58315218715234647339268454366, −0.56836580544248936518399511761, −0.51056899168246572856474907228, −0.44428843303494600532296142044, −0.43121461283019072398784092311, −0.21500873562656545217412207866, −0.19830076886808140690456727962, −0.19684317310263933227371416600, 0.19684317310263933227371416600, 0.19830076886808140690456727962, 0.21500873562656545217412207866, 0.43121461283019072398784092311, 0.44428843303494600532296142044, 0.51056899168246572856474907228, 0.56836580544248936518399511761, 0.58315218715234647339268454366, 0.61121635984475864695812019005, 0.76765634995372156350424394191, 0.818601724168504721975599420571, 0.846091326562409724932860429908, 0.915251977458336363664003567100, 0.915823152696376132682575643525, 0.923927910640083246920424549825, 0.925191814483574197454031239971, 0.936401792492719491646624141885, 0.999606620217287016226190991988, 1.02102102930052395706366641683, 1.14385158479943065221687485645, 1.14943795350724557058164068012, 1.15005556925190022366101250321, 1.21377494398929273802401269561, 1.25805989345304742219997418230, 1.26369159960921172172241333135

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

Plot not available for L-functions of degree greater than 10.