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 − 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 + 139-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 − 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 + 0.0848·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.07533\times 10^{22}\) |
| Root analytic conductor: | \(3.69518\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{40} \cdot 5^{20} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3416082527\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3416082527\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{4} )^{5} \) |
| 3 | \( 1 \) | |
| 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 | \( 1 - 58 T^{2} + 2229 T^{4} - 67576 T^{6} + 1658546 T^{8} - 34351324 T^{10} + 1658546 p^{2} T^{12} - 67576 p^{4} T^{14} + 2229 p^{6} T^{16} - 58 p^{8} T^{18} + p^{10} T^{20} \) | |
| good | 7 | \( ( 1 + 2 T + 2 T^{2} + 22 T^{3} + 69 T^{4} + 64 T^{5} + 232 T^{6} + 912 T^{7} + 4218 T^{8} + 9332 T^{9} + 16332 T^{10} + 9332 p T^{11} + 4218 p^{2} T^{12} + 912 p^{3} T^{13} + 232 p^{4} T^{14} + 64 p^{5} T^{15} + 69 p^{6} T^{16} + 22 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) |
| 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
−1.84307664942865687205695623814, −1.83314722742790328978543828152, −1.82680310379471601842830446613, −1.76889119091652847875617514893, −1.68423575356467772194834956685, −1.67178319337808354709593806692, −1.63680926923515856458952302684, −1.49575940487020992931202238431, −1.48952595214110529636996427165, −1.33113088210507698080741066215, −1.25654858657748579906964201179, −1.19431185271024043385234688983, −1.05747703714534562030669056503, −0.977675735365291501020147926977, −0.977503926332904152289526480557, −0.946322627973314659724928125183, −0.881091391610153892628912436292, −0.77116111885728919167118354560, −0.70042307782204370039495096567, −0.56896372286145013399843839109, −0.44116787071862627035374925156, −0.25308652899120691770597276131, −0.22966079033698631933562749785, −0.15839025031149356573316980429, −0.097557863576593865631639030563, 0.097557863576593865631639030563, 0.15839025031149356573316980429, 0.22966079033698631933562749785, 0.25308652899120691770597276131, 0.44116787071862627035374925156, 0.56896372286145013399843839109, 0.70042307782204370039495096567, 0.77116111885728919167118354560, 0.881091391610153892628912436292, 0.946322627973314659724928125183, 0.977503926332904152289526480557, 0.977675735365291501020147926977, 1.05747703714534562030669056503, 1.19431185271024043385234688983, 1.25654858657748579906964201179, 1.33113088210507698080741066215, 1.48952595214110529636996427165, 1.49575940487020992931202238431, 1.63680926923515856458952302684, 1.67178319337808354709593806692, 1.68423575356467772194834956685, 1.76889119091652847875617514893, 1.82680310379471601842830446613, 1.83314722742790328978543828152, 1.84307664942865687205695623814
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.