Properties

Label 40-570e20-1.1-c1e20-0-0
Degree $40$
Conductor $1.311\times 10^{55}$
Sign $1$
Analytic cond. $1.45559\times 10^{13}$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·5-s − 4·7-s − 8·11-s − 5·16-s − 12·17-s − 4·23-s + 58·25-s − 48·35-s − 12·43-s − 44·47-s + 8·49-s − 96·55-s − 4·73-s + 32·77-s − 60·80-s − 5·81-s + 76·83-s − 144·85-s − 80·101-s + 20·112-s − 48·115-s + 48·119-s − 36·121-s + 124·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 5.36·5-s − 1.51·7-s − 2.41·11-s − 5/4·16-s − 2.91·17-s − 0.834·23-s + 58/5·25-s − 8.11·35-s − 1.82·43-s − 6.41·47-s + 8/7·49-s − 12.9·55-s − 0.468·73-s + 3.64·77-s − 6.70·80-s − 5/9·81-s + 8.34·83-s − 15.6·85-s − 7.96·101-s + 1.88·112-s − 4.47·115-s + 4.40·119-s − 3.27·121-s + 11.0·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20}\)
Sign: $1$
Analytic conductor: \(1.45559\times 10^{13}\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.0001800708223\)
\(L(\frac12)\) \(\approx\) \(0.0001800708223\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{5} \)
3 \( ( 1 + T^{4} )^{5} \)
5 \( ( 1 - 6 T + p^{2} T^{2} - 16 p T^{3} + 206 T^{4} - 492 T^{5} + 206 p T^{6} - 16 p^{3} T^{7} + p^{5} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( 1 - 58 T^{2} + 2229 T^{4} - 67576 T^{6} + 1658546 T^{8} - 34351324 T^{10} + 1658546 p^{2} T^{12} - 67576 p^{4} T^{14} + 2229 p^{6} T^{16} - 58 p^{8} T^{18} + p^{10} T^{20} \)
good7 \( ( 1 + 2 T + 2 T^{2} + 22 T^{3} + 69 T^{4} + 64 T^{5} + 232 T^{6} + 912 T^{7} + 4218 T^{8} + 9332 T^{9} + 16332 T^{10} + 9332 p T^{11} + 4218 p^{2} T^{12} + 912 p^{3} T^{13} + 232 p^{4} T^{14} + 64 p^{5} T^{15} + 69 p^{6} T^{16} + 22 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
11 \( ( 1 + 2 T + 19 T^{2} + 72 T^{3} + 382 T^{4} + 700 T^{5} + 382 p T^{6} + 72 p^{2} T^{7} + 19 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
13 \( 1 + 18 p T^{4} + 14813 T^{8} + 5804984 T^{12} + 843835058 T^{16} - 15430550212 T^{20} + 843835058 p^{4} T^{24} + 5804984 p^{8} T^{28} + 14813 p^{12} T^{32} + 18 p^{17} T^{36} + p^{20} T^{40} \)
17 \( ( 1 + 6 T + 18 T^{2} + 38 T^{3} - 11 p T^{4} - 40 T^{5} + 3848 T^{6} + 35288 T^{7} + 65386 T^{8} - 529564 T^{9} - 2293012 T^{10} - 529564 p T^{11} + 65386 p^{2} T^{12} + 35288 p^{3} T^{13} + 3848 p^{4} T^{14} - 40 p^{5} T^{15} - 11 p^{7} T^{16} + 38 p^{7} T^{17} + 18 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
23 \( ( 1 + 2 T + 2 T^{2} + 22 T^{3} - 571 T^{4} - 128 T^{5} + 1128 T^{6} + 49424 T^{7} + 41818 T^{8} - 1949036 T^{9} - 2242100 T^{10} - 1949036 p T^{11} + 41818 p^{2} T^{12} + 49424 p^{3} T^{13} + 1128 p^{4} T^{14} - 128 p^{5} T^{15} - 571 p^{6} T^{16} + 22 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
29 \( ( 1 + 178 T^{2} + 16181 T^{4} + 977816 T^{6} + 42954834 T^{8} + 1425149740 T^{10} + 42954834 p^{2} T^{12} + 977816 p^{4} T^{14} + 16181 p^{6} T^{16} + 178 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
31 \( ( 1 - 118 T^{2} + 8173 T^{4} - 456 p^{2} T^{6} + 18391570 T^{8} - 624092420 T^{10} + 18391570 p^{2} T^{12} - 456 p^{6} T^{14} + 8173 p^{6} T^{16} - 118 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
37 \( 1 - 1654 T^{4} - 2767171 T^{8} + 6519747128 T^{12} + 3783857540402 T^{16} - 17705976239127044 T^{20} + 3783857540402 p^{4} T^{24} + 6519747128 p^{8} T^{28} - 2767171 p^{12} T^{32} - 1654 p^{16} T^{36} + p^{20} T^{40} \)
41 \( ( 1 - 346 T^{2} + 55973 T^{4} - 5571560 T^{6} + 377863674 T^{8} - 18244901404 T^{10} + 377863674 p^{2} T^{12} - 5571560 p^{4} T^{14} + 55973 p^{6} T^{16} - 346 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
43 \( ( 1 + 6 T + 18 T^{2} + 482 T^{3} + 2773 T^{4} - 7640 T^{5} + 20408 T^{6} + 266872 T^{7} - 8146654 T^{8} - 44140924 T^{9} - 16980052 T^{10} - 44140924 p T^{11} - 8146654 p^{2} T^{12} + 266872 p^{3} T^{13} + 20408 p^{4} T^{14} - 7640 p^{5} T^{15} + 2773 p^{6} T^{16} + 482 p^{7} T^{17} + 18 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
47 \( ( 1 + 22 T + 242 T^{2} + 2002 T^{3} + 15509 T^{4} + 125424 T^{5} + 1010152 T^{6} + 7651552 T^{7} + 55728538 T^{8} + 396981932 T^{9} + 2764845932 T^{10} + 396981932 p T^{11} + 55728538 p^{2} T^{12} + 7651552 p^{3} T^{13} + 1010152 p^{4} T^{14} + 125424 p^{5} T^{15} + 15509 p^{6} T^{16} + 2002 p^{7} T^{17} + 242 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
53 \( 1 - 2886 T^{4} + 29503133 T^{8} - 61528974536 T^{12} + 393256699990418 T^{16} - 637815189733069412 T^{20} + 393256699990418 p^{4} T^{24} - 61528974536 p^{8} T^{28} + 29503133 p^{12} T^{32} - 2886 p^{16} T^{36} + p^{20} T^{40} \)
59 \( ( 1 + 430 T^{2} + 83069 T^{4} + 9730808 T^{6} + 804086762 T^{8} + 52266523860 T^{10} + 804086762 p^{2} T^{12} + 9730808 p^{4} T^{14} + 83069 p^{6} T^{16} + 430 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 + 57 T^{2} - 816 T^{3} + 4226 T^{4} - 43552 T^{5} + 4226 p T^{6} - 816 p^{2} T^{7} + 57 p^{3} T^{8} + p^{5} T^{10} )^{4} \)
67 \( 1 - 11366 T^{4} + 58818365 T^{8} - 109787309896 T^{12} - 587089228812270 T^{16} + 5313462226296719836 T^{20} - 587089228812270 p^{4} T^{24} - 109787309896 p^{8} T^{28} + 58818365 p^{12} T^{32} - 11366 p^{16} T^{36} + p^{20} T^{40} \)
71 \( ( 1 - 510 T^{2} + 125533 T^{4} - 19720936 T^{6} + 2192257362 T^{8} - 180018619060 T^{10} + 2192257362 p^{2} T^{12} - 19720936 p^{4} T^{14} + 125533 p^{6} T^{16} - 510 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
73 \( ( 1 + 2 T + 2 T^{2} - 1166 T^{3} - 5427 T^{4} + 20248 T^{5} + 731128 T^{6} + 4429848 T^{7} + 17562626 T^{8} - 483345364 T^{9} - 1994196340 T^{10} - 483345364 p T^{11} + 17562626 p^{2} T^{12} + 4429848 p^{3} T^{13} + 731128 p^{4} T^{14} + 20248 p^{5} T^{15} - 5427 p^{6} T^{16} - 1166 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
79 \( ( 1 + 406 T^{2} + 66413 T^{4} + 4902920 T^{6} + 51068722 T^{8} - 13116852924 T^{10} + 51068722 p^{2} T^{12} + 4902920 p^{4} T^{14} + 66413 p^{6} T^{16} + 406 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
83 \( ( 1 - 38 T + 722 T^{2} - 11298 T^{3} + 165133 T^{4} - 2018952 T^{5} + 21316552 T^{6} - 216314808 T^{7} + 2092731946 T^{8} - 18902488948 T^{9} + 169284223532 T^{10} - 18902488948 p T^{11} + 2092731946 p^{2} T^{12} - 216314808 p^{3} T^{13} + 21316552 p^{4} T^{14} - 2018952 p^{5} T^{15} + 165133 p^{6} T^{16} - 11298 p^{7} T^{17} + 722 p^{8} T^{18} - 38 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
89 \( ( 1 + 698 T^{2} + 232421 T^{4} + 48436136 T^{6} + 6982269594 T^{8} + 726794172380 T^{10} + 6982269594 p^{2} T^{12} + 48436136 p^{4} T^{14} + 232421 p^{6} T^{16} + 698 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
97 \( 1 - 14166 T^{4} + 125705005 T^{8} - 882571469576 T^{12} + 18771361715902290 T^{16} - \)\(23\!\cdots\!44\)\( T^{20} + 18771361715902290 p^{4} T^{24} - 882571469576 p^{8} T^{28} + 125705005 p^{12} T^{32} - 14166 p^{16} T^{36} + p^{20} T^{40} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.31292943802270084438683651376, −2.28413096497609715801058584050, −2.27443804724121211961452382716, −2.27439687531762654820476355818, −2.26561377250098940332346394833, −2.17131990697141187011829623520, −2.11436836763297951496611674161, −2.05716475673625575243449060031, −2.03997196886770508374899454821, −1.93730779021921061175212371712, −1.85229659862933642519376542140, −1.57866209089037515794718852425, −1.54337266397143508784882773733, −1.49295005579740044072137870860, −1.45336961431069681295976527545, −1.40220863862560393811927972512, −1.32761588743658772649052663833, −1.22170529453993031958503762319, −1.20898469403615763449748684487, −0.964224966801169211451947229202, −0.74186747329499734012610079342, −0.50595066137470423855929677772, −0.26574610698723918410087528684, −0.14336639912651409025233632331, −0.00263088369157415195831078720, 0.00263088369157415195831078720, 0.14336639912651409025233632331, 0.26574610698723918410087528684, 0.50595066137470423855929677772, 0.74186747329499734012610079342, 0.964224966801169211451947229202, 1.20898469403615763449748684487, 1.22170529453993031958503762319, 1.32761588743658772649052663833, 1.40220863862560393811927972512, 1.45336961431069681295976527545, 1.49295005579740044072137870860, 1.54337266397143508784882773733, 1.57866209089037515794718852425, 1.85229659862933642519376542140, 1.93730779021921061175212371712, 2.03997196886770508374899454821, 2.05716475673625575243449060031, 2.11436836763297951496611674161, 2.17131990697141187011829623520, 2.26561377250098940332346394833, 2.27439687531762654820476355818, 2.27443804724121211961452382716, 2.28413096497609715801058584050, 2.31292943802270084438683651376

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

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