Properties

Label 20-1040e10-1.1-c1e10-0-2
Degree $20$
Conductor $1.480\times 10^{30}$
Sign $1$
Analytic cond. $1.55992\times 10^{9}$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 2·7-s + 11·9-s + 2·13-s + 4·25-s + 8·29-s + 6·35-s − 22·37-s + 33·45-s − 2·47-s − 27·49-s − 16·61-s + 22·63-s + 6·65-s − 16·67-s − 4·73-s − 28·79-s + 60·81-s + 4·83-s + 4·91-s + 72·97-s + 32·101-s + 22·117-s + 46·121-s + 9·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.755·7-s + 11/3·9-s + 0.554·13-s + 4/5·25-s + 1.48·29-s + 1.01·35-s − 3.61·37-s + 4.91·45-s − 0.291·47-s − 3.85·49-s − 2.04·61-s + 2.77·63-s + 0.744·65-s − 1.95·67-s − 0.468·73-s − 3.15·79-s + 20/3·81-s + 0.439·83-s + 0.419·91-s + 7.31·97-s + 3.18·101-s + 2.03·117-s + 4.18·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{40} \cdot 5^{10} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(1.55992\times 10^{9}\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{40} \cdot 5^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(34.89643909\)
\(L(\frac12)\) \(\approx\) \(34.89643909\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} - 12 T^{3} + 14 T^{4} + 14 T^{5} + 14 p T^{6} - 12 p^{2} T^{7} + p^{4} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
13 \( 1 - 2 T - 7 T^{2} + 40 T^{3} + 50 T^{4} - 1132 T^{5} + 50 p T^{6} + 40 p^{2} T^{7} - 7 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
good3 \( 1 - 11 T^{2} + 61 T^{4} - 212 T^{6} + 554 T^{8} - 1442 T^{10} + 554 p^{2} T^{12} - 212 p^{4} T^{14} + 61 p^{6} T^{16} - 11 p^{8} T^{18} + p^{10} T^{20} \)
7 \( ( 1 - T + 15 T^{2} - 12 T^{3} + 134 T^{4} - 134 T^{5} + 134 p T^{6} - 12 p^{2} T^{7} + 15 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
11 \( 1 - 46 T^{2} + 965 T^{4} - 13192 T^{6} + 147954 T^{8} - 1605460 T^{10} + 147954 p^{2} T^{12} - 13192 p^{4} T^{14} + 965 p^{6} T^{16} - 46 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 63 T^{2} + 2517 T^{4} - 72708 T^{6} + 1654922 T^{8} - 31028090 T^{10} + 1654922 p^{2} T^{12} - 72708 p^{4} T^{14} + 2517 p^{6} T^{16} - 63 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 - 98 T^{2} + 4949 T^{4} - 175160 T^{6} + 4736210 T^{8} - 100809868 T^{10} + 4736210 p^{2} T^{12} - 175160 p^{4} T^{14} + 4949 p^{6} T^{16} - 98 p^{8} T^{18} + p^{10} T^{20} \)
23 \( 1 - 186 T^{2} + 16317 T^{4} - 889304 T^{6} + 33346578 T^{8} - 898888412 T^{10} + 33346578 p^{2} T^{12} - 889304 p^{4} T^{14} + 16317 p^{6} T^{16} - 186 p^{8} T^{18} + p^{10} T^{20} \)
29 \( ( 1 - 4 T + 89 T^{2} - 144 T^{3} + 3234 T^{4} - 1688 T^{5} + 3234 p T^{6} - 144 p^{2} T^{7} + 89 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( 1 - 202 T^{2} + 18957 T^{4} - 1128280 T^{6} + 49146482 T^{8} - 1692179260 T^{10} + 49146482 p^{2} T^{12} - 1128280 p^{4} T^{14} + 18957 p^{6} T^{16} - 202 p^{8} T^{18} + p^{10} T^{20} \)
37 \( ( 1 + 11 T + 213 T^{2} + 1596 T^{3} + 16670 T^{4} + 88002 T^{5} + 16670 p T^{6} + 1596 p^{2} T^{7} + 213 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
41 \( 1 - 202 T^{2} + 21245 T^{4} - 1520632 T^{6} + 83484562 T^{8} - 3747733244 T^{10} + 83484562 p^{2} T^{12} - 1520632 p^{4} T^{14} + 21245 p^{6} T^{16} - 202 p^{8} T^{18} + p^{10} T^{20} \)
43 \( 1 - 219 T^{2} + 26861 T^{4} - 2252564 T^{6} + 141711914 T^{8} - 6884853698 T^{10} + 141711914 p^{2} T^{12} - 2252564 p^{4} T^{14} + 26861 p^{6} T^{16} - 219 p^{8} T^{18} + p^{10} T^{20} \)
47 \( ( 1 + T + 215 T^{2} + 172 T^{3} + 19334 T^{4} + 11814 T^{5} + 19334 p T^{6} + 172 p^{2} T^{7} + 215 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
53 \( 1 - 166 T^{2} + 19301 T^{4} - 1717192 T^{6} + 122805394 T^{8} - 7039697252 T^{10} + 122805394 p^{2} T^{12} - 1717192 p^{4} T^{14} + 19301 p^{6} T^{16} - 166 p^{8} T^{18} + p^{10} T^{20} \)
59 \( 1 - 274 T^{2} + 40005 T^{4} - 4124152 T^{6} + 333328658 T^{8} - 21783115564 T^{10} + 333328658 p^{2} T^{12} - 4124152 p^{4} T^{14} + 40005 p^{6} T^{16} - 274 p^{8} T^{18} + p^{10} T^{20} \)
61 \( ( 1 + 8 T + 185 T^{2} + 1280 T^{3} + 14562 T^{4} + 96752 T^{5} + 14562 p T^{6} + 1280 p^{2} T^{7} + 185 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( ( 1 + 8 T + 255 T^{2} + 1440 T^{3} + 29642 T^{4} + 131632 T^{5} + 29642 p T^{6} + 1440 p^{2} T^{7} + 255 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
71 \( 1 - 235 T^{2} + 27925 T^{4} - 2415092 T^{6} + 207629498 T^{8} - 16205410050 T^{10} + 207629498 p^{2} T^{12} - 2415092 p^{4} T^{14} + 27925 p^{6} T^{16} - 235 p^{8} T^{18} + p^{10} T^{20} \)
73 \( ( 1 + 2 T + 261 T^{2} + 696 T^{3} + 33026 T^{4} + 73548 T^{5} + 33026 p T^{6} + 696 p^{2} T^{7} + 261 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
79 \( ( 1 + 14 T + 251 T^{2} + 2248 T^{3} + 33402 T^{4} + 264404 T^{5} + 33402 p T^{6} + 2248 p^{2} T^{7} + 251 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( ( 1 - 2 T + 103 T^{2} - 1240 T^{3} + 18018 T^{4} - 45548 T^{5} + 18018 p T^{6} - 1240 p^{2} T^{7} + 103 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 - 366 T^{2} + 71421 T^{4} - 10162664 T^{6} + 1185209586 T^{8} - 115893586388 T^{10} + 1185209586 p^{2} T^{12} - 10162664 p^{4} T^{14} + 71421 p^{6} T^{16} - 366 p^{8} T^{18} + p^{10} T^{20} \)
97 \( ( 1 - 36 T + 901 T^{2} - 160 p T^{3} + 214618 T^{4} - 2328696 T^{5} + 214618 p T^{6} - 160 p^{3} T^{7} + 901 p^{3} T^{8} - 36 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.53873025416977412005105642745, −3.45400299421533641320898490759, −3.44317526548194928790469740436, −3.27727635828882598414171702766, −3.07786339284439538302207026765, −3.05904735142757547288927705272, −3.05128006368634466300430933463, −2.92889627297938234199895610417, −2.90203556056705662347888427899, −2.45739461101794668605610897778, −2.41305728711922068985552708235, −2.16123510338490939759453734421, −1.99045470697162462484057264141, −1.96726787009000147429073562949, −1.87730707988957579848251393727, −1.82904635834694899251108962025, −1.62861498094427358736589352617, −1.53586165455080469020649513805, −1.48415001492905069308488050465, −1.46736168650280227653785685243, −1.24055711334998050469884817195, −0.72559926813134668166592820908, −0.68504282224356816901857194584, −0.62203956335964590264804857224, −0.39863465835384188878208771331, 0.39863465835384188878208771331, 0.62203956335964590264804857224, 0.68504282224356816901857194584, 0.72559926813134668166592820908, 1.24055711334998050469884817195, 1.46736168650280227653785685243, 1.48415001492905069308488050465, 1.53586165455080469020649513805, 1.62861498094427358736589352617, 1.82904635834694899251108962025, 1.87730707988957579848251393727, 1.96726787009000147429073562949, 1.99045470697162462484057264141, 2.16123510338490939759453734421, 2.41305728711922068985552708235, 2.45739461101794668605610897778, 2.90203556056705662347888427899, 2.92889627297938234199895610417, 3.05128006368634466300430933463, 3.05904735142757547288927705272, 3.07786339284439538302207026765, 3.27727635828882598414171702766, 3.44317526548194928790469740436, 3.45400299421533641320898490759, 3.53873025416977412005105642745

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

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