Properties

Label 40-570e20-1.1-c1e20-0-2
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

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 4·7-s − 8·11-s − 5·16-s + 4·17-s + 44·23-s + 10·25-s + 16·35-s + 52·43-s + 4·47-s + 8·49-s + 32·55-s + 32·61-s − 20·73-s + 32·77-s + 20·80-s − 5·81-s − 116·83-s − 16·85-s + 16·101-s + 20·112-s − 176·115-s − 16·119-s − 52·121-s − 36·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.51·7-s − 2.41·11-s − 5/4·16-s + 0.970·17-s + 9.17·23-s + 2·25-s + 2.70·35-s + 7.92·43-s + 0.583·47-s + 8/7·49-s + 4.31·55-s + 4.09·61-s − 2.34·73-s + 3.64·77-s + 2.23·80-s − 5/9·81-s − 12.7·83-s − 1.73·85-s + 1.59·101-s + 1.88·112-s − 16.4·115-s − 1.46·119-s − 4.72·121-s − 3.21·125-s + 0.0887·127-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(5.179132307\)
\(L(\frac12)\) \(\approx\) \(5.179132307\)
\(L(\frac{3}{2})\) 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 \( ( 1 + 2 T + T^{2} + 8 T^{3} + 22 T^{4} + 52 T^{5} + 22 p T^{6} + 8 p^{2} T^{7} + p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
19 \( 1 + 54 T^{2} + 1909 T^{4} + 43720 T^{6} + 834034 T^{8} + 14825156 T^{10} + 834034 p^{2} T^{12} + 43720 p^{4} T^{14} + 1909 p^{6} T^{16} + 54 p^{8} T^{18} + p^{10} T^{20} \)
good7 \( ( 1 + 2 T + 2 T^{2} + 22 T^{3} + 5 T^{4} - 32 T^{5} + 24 p T^{6} + 752 T^{7} - 710 T^{8} - 4428 T^{9} + 8780 T^{10} - 4428 p T^{11} - 710 p^{2} T^{12} + 752 p^{3} T^{13} + 24 p^{5} T^{14} - 32 p^{5} T^{15} + 5 p^{6} T^{16} + 22 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
11 \( ( 1 + 2 T + 23 T^{2} + 80 T^{3} + 38 p T^{4} + 1004 T^{5} + 38 p^{2} T^{6} + 80 p^{2} T^{7} + 23 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
13 \( 1 + 394 T^{4} + 49245 T^{8} - 310216 T^{12} - 475711438 T^{16} - 36658781444 T^{20} - 475711438 p^{4} T^{24} - 310216 p^{8} T^{28} + 49245 p^{12} T^{32} + 394 p^{16} T^{36} + p^{20} T^{40} \)
17 \( ( 1 - 2 T + 2 T^{2} - 18 T^{3} + 37 T^{4} + 264 T^{5} - 440 T^{6} + 4008 T^{7} + 50186 T^{8} - 92636 T^{9} + 63884 T^{10} - 92636 p T^{11} + 50186 p^{2} T^{12} + 4008 p^{3} T^{13} - 440 p^{4} T^{14} + 264 p^{5} T^{15} + 37 p^{6} T^{16} - 18 p^{7} T^{17} + 2 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
23 \( ( 1 - 22 T + 242 T^{2} - 1970 T^{3} + 14021 T^{4} - 90816 T^{5} + 545320 T^{6} - 3130352 T^{7} + 17263066 T^{8} - 89682716 T^{9} + 439992428 T^{10} - 89682716 p T^{11} + 17263066 p^{2} T^{12} - 3130352 p^{3} T^{13} + 545320 p^{4} T^{14} - 90816 p^{5} T^{15} + 14021 p^{6} T^{16} - 1970 p^{7} T^{17} + 242 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
29 \( ( 1 + 154 T^{2} + 13381 T^{4} + 781592 T^{6} + 33862338 T^{8} + 1116062556 T^{10} + 33862338 p^{2} T^{12} + 781592 p^{4} T^{14} + 13381 p^{6} T^{16} + 154 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
31 \( ( 1 - 174 T^{2} + 16605 T^{4} - 1057928 T^{6} + 49482530 T^{8} - 1750617876 T^{10} + 49482530 p^{2} T^{12} - 1057928 p^{4} T^{14} + 16605 p^{6} T^{16} - 174 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
37 \( 1 - 6870 T^{4} + 19838781 T^{8} - 28798551368 T^{12} + 17864303461554 T^{16} - 4119255381666372 T^{20} + 17864303461554 p^{4} T^{24} - 28798551368 p^{8} T^{28} + 19838781 p^{12} T^{32} - 6870 p^{16} T^{36} + p^{20} T^{40} \)
41 \( ( 1 - 194 T^{2} + 18661 T^{4} - 1177768 T^{6} + 56689978 T^{8} - 2395807948 T^{10} + 56689978 p^{2} T^{12} - 1177768 p^{4} T^{14} + 18661 p^{6} T^{16} - 194 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
43 \( ( 1 - 26 T + 338 T^{2} - 3246 T^{3} + 26693 T^{4} - 190376 T^{5} + 1195800 T^{6} - 6457112 T^{7} + 27650866 T^{8} - 104552492 T^{9} + 534178924 T^{10} - 104552492 p T^{11} + 27650866 p^{2} T^{12} - 6457112 p^{3} T^{13} + 1195800 p^{4} T^{14} - 190376 p^{5} T^{15} + 26693 p^{6} T^{16} - 3246 p^{7} T^{17} + 338 p^{8} T^{18} - 26 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
47 \( ( 1 - 2 T + 2 T^{2} - 1062 T^{3} + 6165 T^{4} + 5056 T^{5} + 541480 T^{6} - 5016464 T^{7} + 4811418 T^{8} - 120756084 T^{9} + 2233931468 T^{10} - 120756084 p T^{11} + 4811418 p^{2} T^{12} - 5016464 p^{3} T^{13} + 541480 p^{4} T^{14} + 5056 p^{5} T^{15} + 6165 p^{6} T^{16} - 1062 p^{7} T^{17} + 2 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
53 \( 1 - 3718 T^{4} + 21469341 T^{8} - 41973520328 T^{12} + 221674170955666 T^{16} - 405771832358172132 T^{20} + 221674170955666 p^{4} T^{24} - 41973520328 p^{8} T^{28} + 21469341 p^{12} T^{32} - 3718 p^{16} T^{36} + p^{20} T^{40} \)
59 \( ( 1 + 446 T^{2} + 94685 T^{4} + 12630840 T^{6} + 1177962250 T^{8} + 80575224628 T^{10} + 1177962250 p^{2} T^{12} + 12630840 p^{4} T^{14} + 94685 p^{6} T^{16} + 446 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 - 8 T + 289 T^{2} - 1840 T^{3} + 34362 T^{4} - 165264 T^{5} + 34362 p T^{6} - 1840 p^{2} T^{7} + 289 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
67 \( 1 - 5990 T^{4} - 25704643 T^{8} + 271551620280 T^{12} + 202002443610130 T^{16} - 6356983071451297828 T^{20} + 202002443610130 p^{4} T^{24} + 271551620280 p^{8} T^{28} - 25704643 p^{12} T^{32} - 5990 p^{16} T^{36} + p^{20} T^{40} \)
71 \( ( 1 - 478 T^{2} + 114525 T^{4} - 17788776 T^{6} + 1967920210 T^{8} - 161370485236 T^{10} + 1967920210 p^{2} T^{12} - 17788776 p^{4} T^{14} + 114525 p^{6} T^{16} - 478 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
73 \( ( 1 + 10 T + 50 T^{2} + 1434 T^{3} + 15373 T^{4} - 13928 T^{5} + 120248 T^{6} + 2848472 T^{7} - 108763454 T^{8} - 1043040772 T^{9} - 2659104084 T^{10} - 1043040772 p T^{11} - 108763454 p^{2} T^{12} + 2848472 p^{3} T^{13} + 120248 p^{4} T^{14} - 13928 p^{5} T^{15} + 15373 p^{6} T^{16} + 1434 p^{7} T^{17} + 50 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
79 \( ( 1 + 606 T^{2} + 174525 T^{4} + 31601096 T^{6} + 3989973218 T^{8} + 367012757556 T^{10} + 3989973218 p^{2} T^{12} + 31601096 p^{4} T^{14} + 174525 p^{6} T^{16} + 606 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
83 \( ( 1 + 58 T + 1682 T^{2} + 34590 T^{3} + 586861 T^{4} + 8668952 T^{5} + 113933064 T^{6} + 1361047912 T^{7} + 14963593546 T^{8} + 151896077932 T^{9} + 1432347968812 T^{10} + 151896077932 p T^{11} + 14963593546 p^{2} T^{12} + 1361047912 p^{3} T^{13} + 113933064 p^{4} T^{14} + 8668952 p^{5} T^{15} + 586861 p^{6} T^{16} + 34590 p^{7} T^{17} + 1682 p^{8} T^{18} + 58 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
89 \( ( 1 + 418 T^{2} + 83941 T^{4} + 10954856 T^{6} + 1087940634 T^{8} + 97111095180 T^{10} + 1087940634 p^{2} T^{12} + 10954856 p^{4} T^{14} + 83941 p^{6} T^{16} + 418 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
97 \( 1 - 30614 T^{4} + 404194349 T^{8} - 1846939408392 T^{12} - 21181414394442158 T^{16} + \)\(39\!\cdots\!16\)\( T^{20} - 21181414394442158 p^{4} T^{24} - 1846939408392 p^{8} T^{28} + 404194349 p^{12} T^{32} - 30614 p^{16} T^{36} + p^{20} T^{40} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.57735728379683710846267350506, −2.56815719386129822675360563184, −2.51145575258180944356449347853, −2.32909964388104243270299691597, −2.32003951494753541235983530695, −2.23975493599900291725633329393, −2.20155277323086302692439126398, −2.10522909792603599699057653833, −1.76703155740687037480109764950, −1.75237792292057775905766191176, −1.66383942899470364336610096666, −1.60803696079533543218176037815, −1.51554366828672351667668815223, −1.31806937097873496580790855478, −1.27964757098263930999947532782, −1.26132246646093364993942878734, −1.13083234475132818666413665332, −1.06443692025271496493354410452, −0.988382596465890019942722416085, −0.817992116117838225306405951640, −0.75226089350219586088090635219, −0.55008891708457845029654201241, −0.54232341356013299088050799064, −0.26957826551861490946452591371, −0.23468687304048084947566864387, 0.23468687304048084947566864387, 0.26957826551861490946452591371, 0.54232341356013299088050799064, 0.55008891708457845029654201241, 0.75226089350219586088090635219, 0.817992116117838225306405951640, 0.988382596465890019942722416085, 1.06443692025271496493354410452, 1.13083234475132818666413665332, 1.26132246646093364993942878734, 1.27964757098263930999947532782, 1.31806937097873496580790855478, 1.51554366828672351667668815223, 1.60803696079533543218176037815, 1.66383942899470364336610096666, 1.75237792292057775905766191176, 1.76703155740687037480109764950, 2.10522909792603599699057653833, 2.20155277323086302692439126398, 2.23975493599900291725633329393, 2.32003951494753541235983530695, 2.32909964388104243270299691597, 2.51145575258180944356449347853, 2.56815719386129822675360563184, 2.57735728379683710846267350506

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

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