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Properties

Label 20-2900e10-1.1-c1e10-0-0
Degree $20$
Conductor $4.207\times 10^{34}$
Sign $1$
Analytic cond. $4.43354\times 10^{13}$
Root an. cond. $4.81213$
Motivic weight $1$
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  + 14·9-s − 4·13-s + 14·23-s + 2·29-s − 18·49-s + 4·53-s − 6·59-s − 22·67-s + 14·71-s + 98·81-s + 6·83-s + 20·103-s − 6·107-s − 56·117-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 54·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 14/3·9-s − 1.10·13-s + 2.91·23-s + 0.371·29-s − 2.57·49-s + 0.549·53-s − 0.781·59-s − 2.68·67-s + 1.66·71-s + 98/9·81-s + 0.658·83-s + 1.97·103-s − 0.580·107-s − 5.17·117-s + 4.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{20} \cdot 5^{20} \cdot 29^{10}\)
Sign: $1$
Analytic conductor: \(4.43354\times 10^{13}\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{20} \cdot 5^{20} \cdot 29^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.278884098\)
\(L(\frac12)\) \(\approx\) \(1.278884098\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 - 2 T - 35 T^{2} + 344 T^{3} + 894 T^{4} - 12556 T^{5} + 894 p T^{6} + 344 p^{2} T^{7} - 35 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
good3 \( 1 - 14 T^{2} + 98 T^{4} - 479 T^{6} + 1894 T^{8} - 2081 p T^{10} + 1894 p^{2} T^{12} - 479 p^{4} T^{14} + 98 p^{6} T^{16} - 14 p^{8} T^{18} + p^{10} T^{20} \)
7 \( ( 1 + 9 T^{2} + 9 T^{3} + 18 T^{4} + 113 T^{5} + 18 p T^{6} + 9 p^{2} T^{7} + 9 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
11 \( 1 - 46 T^{2} + 1212 T^{4} - 22114 T^{6} + 318277 T^{8} - 3782811 T^{10} + 318277 p^{2} T^{12} - 22114 p^{4} T^{14} + 1212 p^{6} T^{16} - 46 p^{8} T^{18} + p^{10} T^{20} \)
13 \( ( 1 + 2 T + 33 T^{2} - 17 T^{3} + 310 T^{4} - 1163 T^{5} + 310 p T^{6} - 17 p^{2} T^{7} + 33 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
17 \( 1 - 115 T^{2} + 6400 T^{4} - 13517 p T^{6} + 5933599 T^{8} - 115440383 T^{10} + 5933599 p^{2} T^{12} - 13517 p^{5} T^{14} + 6400 p^{6} T^{16} - 115 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 - 122 T^{2} + 6410 T^{4} - 191611 T^{6} + 3868240 T^{8} - 69735943 T^{10} + 3868240 p^{2} T^{12} - 191611 p^{4} T^{14} + 6410 p^{6} T^{16} - 122 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 - 7 T + 85 T^{2} - 351 T^{3} + 2601 T^{4} - 8359 T^{5} + 2601 p T^{6} - 351 p^{2} T^{7} + 85 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( 1 - 85 T^{2} + 6211 T^{4} - 271409 T^{6} + 11275253 T^{8} - 11209063 p T^{10} + 11275253 p^{2} T^{12} - 271409 p^{4} T^{14} + 6211 p^{6} T^{16} - 85 p^{8} T^{18} + p^{10} T^{20} \)
37 \( 1 - 218 T^{2} + 23494 T^{4} - 1655105 T^{6} + 86131750 T^{8} - 3543132811 T^{10} + 86131750 p^{2} T^{12} - 1655105 p^{4} T^{14} + 23494 p^{6} T^{16} - 218 p^{8} T^{18} + p^{10} T^{20} \)
41 \( 1 - 85 T^{2} + 6160 T^{4} - 301684 T^{6} + 374948 p T^{8} - 604795295 T^{10} + 374948 p^{3} T^{12} - 301684 p^{4} T^{14} + 6160 p^{6} T^{16} - 85 p^{8} T^{18} + p^{10} T^{20} \)
43 \( 1 - 228 T^{2} + 24588 T^{4} - 1702920 T^{6} + 88726491 T^{8} - 3977829607 T^{10} + 88726491 p^{2} T^{12} - 1702920 p^{4} T^{14} + 24588 p^{6} T^{16} - 228 p^{8} T^{18} + p^{10} T^{20} \)
47 \( 1 - 253 T^{2} + 33780 T^{4} - 3076456 T^{6} + 209880354 T^{8} - 11135135751 T^{10} + 209880354 p^{2} T^{12} - 3076456 p^{4} T^{14} + 33780 p^{6} T^{16} - 253 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 - 2 T + 154 T^{2} - 419 T^{3} + 10684 T^{4} - 33233 T^{5} + 10684 p T^{6} - 419 p^{2} T^{7} + 154 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( ( 1 + 3 T + 151 T^{2} + 117 T^{3} + 12863 T^{4} + 7335 T^{5} + 12863 p T^{6} + 117 p^{2} T^{7} + 151 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( 1 - 137 T^{2} + 10076 T^{4} - 247237 T^{6} - 25459145 T^{8} + 2476891565 T^{10} - 25459145 p^{2} T^{12} - 247237 p^{4} T^{14} + 10076 p^{6} T^{16} - 137 p^{8} T^{18} + p^{10} T^{20} \)
67 \( ( 1 + 11 T + 358 T^{2} + 2918 T^{3} + 49516 T^{4} + 292255 T^{5} + 49516 p T^{6} + 2918 p^{2} T^{7} + 358 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
71 \( ( 1 - 7 T + 263 T^{2} - 1785 T^{3} + 33175 T^{4} - 179539 T^{5} + 33175 p T^{6} - 1785 p^{2} T^{7} + 263 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 410 T^{2} + 88881 T^{4} - 12955863 T^{6} + 1397823678 T^{8} - 115766053773 T^{10} + 1397823678 p^{2} T^{12} - 12955863 p^{4} T^{14} + 88881 p^{6} T^{16} - 410 p^{8} T^{18} + p^{10} T^{20} \)
79 \( 1 - 324 T^{2} + 46817 T^{4} - 4847785 T^{6} + 496248010 T^{8} - 44962100021 T^{10} + 496248010 p^{2} T^{12} - 4847785 p^{4} T^{14} + 46817 p^{6} T^{16} - 324 p^{8} T^{18} + p^{10} T^{20} \)
83 \( ( 1 - 3 T + 254 T^{2} - 1065 T^{3} + 35007 T^{4} - 117881 T^{5} + 35007 p T^{6} - 1065 p^{2} T^{7} + 254 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 - 352 T^{2} + 70021 T^{4} - 9661105 T^{6} + 1057254190 T^{8} - 99506347781 T^{10} + 1057254190 p^{2} T^{12} - 9661105 p^{4} T^{14} + 70021 p^{6} T^{16} - 352 p^{8} T^{18} + p^{10} T^{20} \)
97 \( 1 - 385 T^{2} + 68172 T^{4} - 6266660 T^{6} + 243026114 T^{8} - 1078215243 T^{10} + 243026114 p^{2} T^{12} - 6266660 p^{4} T^{14} + 68172 p^{6} T^{16} - 385 p^{8} T^{18} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.05605244077636120263641686132, −2.94087743504373516178568878337, −2.91881969618578632819640048818, −2.79275991176869723407424657665, −2.71188629140847284142195128658, −2.61680266904952910412209944706, −2.43338247680422574029995057920, −2.28165020484239427937905401592, −2.20558669822419382535933931303, −2.03831456004207421393030524547, −2.02384871769946773308073970200, −1.93001306348390920078357973504, −1.66692508234789368491107554455, −1.65708698641137565448254003208, −1.63743901760062980039792998959, −1.48913752685415720370789974292, −1.47306446352934530273723468727, −1.12650575978334970167454927627, −1.09917336150711836566202684124, −0.962038114200720467425059509257, −0.908129877422534017141237752087, −0.67973772006662614849281545037, −0.64496094143292066322665247905, −0.38097511182328039758233744702, −0.04177558258429008527692657144, 0.04177558258429008527692657144, 0.38097511182328039758233744702, 0.64496094143292066322665247905, 0.67973772006662614849281545037, 0.908129877422534017141237752087, 0.962038114200720467425059509257, 1.09917336150711836566202684124, 1.12650575978334970167454927627, 1.47306446352934530273723468727, 1.48913752685415720370789974292, 1.63743901760062980039792998959, 1.65708698641137565448254003208, 1.66692508234789368491107554455, 1.93001306348390920078357973504, 2.02384871769946773308073970200, 2.03831456004207421393030524547, 2.20558669822419382535933931303, 2.28165020484239427937905401592, 2.43338247680422574029995057920, 2.61680266904952910412209944706, 2.71188629140847284142195128658, 2.79275991176869723407424657665, 2.91881969618578632819640048818, 2.94087743504373516178568878337, 3.05605244077636120263641686132

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

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