Dirichlet series
| L(s) = 1 | + 5·4-s − 2·5-s − 10·9-s − 12·11-s + 10·16-s − 22·19-s − 10·20-s − 25-s − 4·29-s − 16·31-s − 50·36-s + 2·41-s − 60·44-s + 20·45-s + 90·49-s + 24·55-s − 44·59-s − 4·61-s + 5·64-s − 44·71-s − 110·76-s − 4·79-s − 20·80-s + 55·81-s + 2·89-s + 44·95-s + 120·99-s + ⋯ |
| L(s) = 1 | + 5/2·4-s − 0.894·5-s − 3.33·9-s − 3.61·11-s + 5/2·16-s − 5.04·19-s − 2.23·20-s − 1/5·25-s − 0.742·29-s − 2.87·31-s − 8.33·36-s + 0.312·41-s − 9.04·44-s + 2.98·45-s + 90/7·49-s + 3.23·55-s − 5.72·59-s − 0.512·61-s + 5/8·64-s − 5.22·71-s − 12.6·76-s − 0.450·79-s − 2.23·80-s + 55/9·81-s + 0.211·89-s + 4.51·95-s + 12.0·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 5^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4174.59\) |
| Root analytic conductor: | \(1.23172\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 5^{20} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.008294118325\) |
| \(L(\frac12)\) | \(\approx\) | \(0.008294118325\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{5} \) |
| 5 | \( 1 + 2 T + p T^{2} + 38 T^{3} + 43 T^{4} + 56 T^{5} + 322 T^{6} - 424 T^{7} - 1321 T^{8} - 2706 T^{9} - 14611 T^{10} - 2706 p T^{11} - 1321 p^{2} T^{12} - 424 p^{3} T^{13} + 322 p^{4} T^{14} + 56 p^{5} T^{15} + 43 p^{6} T^{16} + 38 p^{7} T^{17} + p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( ( 1 + 11 T + 39 T^{2} + 4 T^{3} - 229 T^{4} - 543 T^{5} - 229 p T^{6} + 4 p^{2} T^{7} + 39 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| good | 3 | \( 1 + 10 T^{2} + 5 p^{2} T^{4} + 142 T^{6} + 352 T^{8} + 422 T^{10} - 425 T^{12} - 2194 T^{14} - 703 p T^{16} + 10940 p T^{18} + 185248 T^{20} + 10940 p^{3} T^{22} - 703 p^{5} T^{24} - 2194 p^{6} T^{26} - 425 p^{8} T^{28} + 422 p^{10} T^{30} + 352 p^{12} T^{32} + 142 p^{14} T^{34} + 5 p^{18} T^{36} + 10 p^{18} T^{38} + p^{20} T^{40} \) |
| 7 | \( ( 1 - 45 T^{2} + 949 T^{4} - 12692 T^{6} + 123862 T^{8} - 958786 T^{10} + 123862 p^{2} T^{12} - 12692 p^{4} T^{14} + 949 p^{6} T^{16} - 45 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 11 | \( ( 1 + 3 T + 7 T^{2} - 4 p T^{3} + 15 T^{4} - 139 T^{5} + 15 p T^{6} - 4 p^{3} T^{7} + 7 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 13 | \( 1 + 70 T^{2} + 2671 T^{4} + 66414 T^{6} + 1143959 T^{8} + 12846836 T^{10} + 57374482 T^{12} - 1122977348 T^{14} - 2558833727 p T^{16} - 543989176558 T^{18} - 7394361658655 T^{20} - 543989176558 p^{2} T^{22} - 2558833727 p^{5} T^{24} - 1122977348 p^{6} T^{26} + 57374482 p^{8} T^{28} + 12846836 p^{10} T^{30} + 1143959 p^{12} T^{32} + 66414 p^{14} T^{34} + 2671 p^{16} T^{36} + 70 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 + 42 T^{2} + 1103 T^{4} + 514 p T^{6} - 119025 T^{8} - 4713924 T^{10} - 12376966 T^{12} + 79567204 p T^{14} + 25014643341 T^{16} + 83918281118 T^{18} - 3280024025511 T^{20} + 83918281118 p^{2} T^{22} + 25014643341 p^{4} T^{24} + 79567204 p^{7} T^{26} - 12376966 p^{8} T^{28} - 4713924 p^{10} T^{30} - 119025 p^{12} T^{32} + 514 p^{15} T^{34} + 1103 p^{16} T^{36} + 42 p^{18} T^{38} + p^{20} T^{40} \) | |
| 23 | \( 1 + 133 T^{2} + 8276 T^{4} + 362129 T^{6} + 14032623 T^{8} + 495658538 T^{10} + 15497678228 T^{12} + 448605375798 T^{14} + 12341985782817 T^{16} + 313485268561039 T^{18} + 7396923564796536 T^{20} + 313485268561039 p^{2} T^{22} + 12341985782817 p^{4} T^{24} + 448605375798 p^{6} T^{26} + 15497678228 p^{8} T^{28} + 495658538 p^{10} T^{30} + 14032623 p^{12} T^{32} + 362129 p^{14} T^{34} + 8276 p^{16} T^{36} + 133 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 + 2 T - 67 T^{2} - 442 T^{3} + 1435 T^{4} + 20216 T^{5} + 22834 T^{6} - 339952 T^{7} - 1218169 T^{8} + 1859718 T^{9} + 19911749 T^{10} + 1859718 p T^{11} - 1218169 p^{2} T^{12} - 339952 p^{3} T^{13} + 22834 p^{4} T^{14} + 20216 p^{5} T^{15} + 1435 p^{6} T^{16} - 442 p^{7} T^{17} - 67 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 + 4 T + 89 T^{2} + 264 T^{3} + 3814 T^{4} + 9932 T^{5} + 3814 p T^{6} + 264 p^{2} T^{7} + 89 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 37 | \( ( 1 - 137 T^{2} + 11861 T^{4} - 725724 T^{6} + 36316222 T^{8} - 1466939818 T^{10} + 36316222 p^{2} T^{12} - 725724 p^{4} T^{14} + 11861 p^{6} T^{16} - 137 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 41 | \( ( 1 - T - 142 T^{2} + 381 T^{3} + 10778 T^{4} - 36851 T^{5} - 523508 T^{6} + 1914709 T^{7} + 19620269 T^{8} - 36313598 T^{9} - 713692780 T^{10} - 36313598 p T^{11} + 19620269 p^{2} T^{12} + 1914709 p^{3} T^{13} - 523508 p^{4} T^{14} - 36851 p^{5} T^{15} + 10778 p^{6} T^{16} + 381 p^{7} T^{17} - 142 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 43 | \( 1 + 254 T^{2} + 30463 T^{4} + 2557350 T^{6} + 187615799 T^{8} + 12754316804 T^{10} + 780314026610 T^{12} + 42857590237484 T^{14} + 2181087301424461 T^{16} + 105009029424103994 T^{18} + 4712854818602369553 T^{20} + 105009029424103994 p^{2} T^{22} + 2181087301424461 p^{4} T^{24} + 42857590237484 p^{6} T^{26} + 780314026610 p^{8} T^{28} + 12754316804 p^{10} T^{30} + 187615799 p^{12} T^{32} + 2557350 p^{14} T^{34} + 30463 p^{16} T^{36} + 254 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 + 230 T^{2} + 30599 T^{4} + 2854638 T^{6} + 204937475 T^{8} + 11903114132 T^{10} + 577386636918 T^{12} + 24276959107420 T^{14} + 928542095856217 T^{16} + 35344275414295514 T^{18} + 1513956131922792185 T^{20} + 35344275414295514 p^{2} T^{22} + 928542095856217 p^{4} T^{24} + 24276959107420 p^{6} T^{26} + 577386636918 p^{8} T^{28} + 11903114132 p^{10} T^{30} + 204937475 p^{12} T^{32} + 2854638 p^{14} T^{34} + 30599 p^{16} T^{36} + 230 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 + 305 T^{2} + 43364 T^{4} + 4345461 T^{6} + 388749779 T^{8} + 31738121438 T^{10} + 2279507853228 T^{12} + 150955127641474 T^{14} + 9508691222412205 T^{16} + 553893234302884427 T^{18} + 30052980726649758680 T^{20} + 553893234302884427 p^{2} T^{22} + 9508691222412205 p^{4} T^{24} + 150955127641474 p^{6} T^{26} + 2279507853228 p^{8} T^{28} + 31738121438 p^{10} T^{30} + 388749779 p^{12} T^{32} + 4345461 p^{14} T^{34} + 43364 p^{16} T^{36} + 305 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( ( 1 + 22 T + 43 T^{2} - 1054 T^{3} + 16226 T^{4} + 215698 T^{5} - 1026365 T^{6} - 5647130 T^{7} + 177757285 T^{8} + 527374480 T^{9} - 7678907312 T^{10} + 527374480 p T^{11} + 177757285 p^{2} T^{12} - 5647130 p^{3} T^{13} - 1026365 p^{4} T^{14} + 215698 p^{5} T^{15} + 16226 p^{6} T^{16} - 1054 p^{7} T^{17} + 43 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 2 T - 139 T^{2} - 378 T^{3} + 6707 T^{4} + 28712 T^{5} - 231510 T^{6} - 39704 p T^{7} + 18299959 T^{8} + 95220830 T^{9} - 1334665051 T^{10} + 95220830 p T^{11} + 18299959 p^{2} T^{12} - 39704 p^{4} T^{13} - 231510 p^{4} T^{14} + 28712 p^{5} T^{15} + 6707 p^{6} T^{16} - 378 p^{7} T^{17} - 139 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 + 298 T^{2} + 47045 T^{4} + 3960958 T^{6} + 97743984 T^{8} - 21663277898 T^{10} - 3152945657937 T^{12} - 198573650897714 T^{14} - 1167230239138029 T^{16} + 954560901302688244 T^{18} + 96409723347846814592 T^{20} + 954560901302688244 p^{2} T^{22} - 1167230239138029 p^{4} T^{24} - 198573650897714 p^{6} T^{26} - 3152945657937 p^{8} T^{28} - 21663277898 p^{10} T^{30} + 97743984 p^{12} T^{32} + 3960958 p^{14} T^{34} + 47045 p^{16} T^{36} + 298 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 + 22 T - 11 T^{2} - 1990 T^{3} + 23603 T^{4} + 370852 T^{5} - 1892726 T^{6} - 14223476 T^{7} + 352957789 T^{8} + 1084313866 T^{9} - 20802571049 T^{10} + 1084313866 p T^{11} + 352957789 p^{2} T^{12} - 14223476 p^{3} T^{13} - 1892726 p^{4} T^{14} + 370852 p^{5} T^{15} + 23603 p^{6} T^{16} - 1990 p^{7} T^{17} - 11 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 + 338 T^{2} + 57521 T^{4} + 7104598 T^{6} + 695423436 T^{8} + 50882576502 T^{10} + 2388366627583 T^{12} - 3611563830674 T^{14} - 16139629743366897 T^{16} - 2130063885214835084 T^{18} - \)\(18\!\cdots\!80\)\( T^{20} - 2130063885214835084 p^{2} T^{22} - 16139629743366897 p^{4} T^{24} - 3611563830674 p^{6} T^{26} + 2388366627583 p^{8} T^{28} + 50882576502 p^{10} T^{30} + 695423436 p^{12} T^{32} + 7104598 p^{14} T^{34} + 57521 p^{16} T^{36} + 338 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 + 2 T - 179 T^{2} - 1346 T^{3} + 15251 T^{4} + 205228 T^{5} + 1142 p T^{6} - 19874684 T^{7} - 132834163 T^{8} + 662451454 T^{9} + 16793738495 T^{10} + 662451454 p T^{11} - 132834163 p^{2} T^{12} - 19874684 p^{3} T^{13} + 1142 p^{5} T^{14} + 205228 p^{5} T^{15} + 15251 p^{6} T^{16} - 1346 p^{7} T^{17} - 179 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 - 558 T^{2} + 151703 T^{4} - 26578980 T^{6} + 3343384121 T^{8} - 317194151954 T^{10} + 3343384121 p^{2} T^{12} - 26578980 p^{4} T^{14} + 151703 p^{6} T^{16} - 558 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 89 | \( ( 1 - T - 272 T^{2} + 727 T^{3} + 33779 T^{4} - 109210 T^{5} - 3626960 T^{6} + 1384910 T^{7} + 430967713 T^{8} + 292358945 T^{9} - 43900040888 T^{10} + 292358945 p T^{11} + 430967713 p^{2} T^{12} + 1384910 p^{3} T^{13} - 3626960 p^{4} T^{14} - 109210 p^{5} T^{15} + 33779 p^{6} T^{16} + 727 p^{7} T^{17} - 272 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 + 462 T^{2} + 96305 T^{4} + 11956010 T^{6} + 1091292780 T^{8} + 111552930858 T^{10} + 15482291389055 T^{12} + 1993553223173138 T^{14} + 194449299271087215 T^{16} + 15108311827632587660 T^{18} + \)\(12\!\cdots\!04\)\( T^{20} + 15108311827632587660 p^{2} T^{22} + 194449299271087215 p^{4} T^{24} + 1993553223173138 p^{6} T^{26} + 15482291389055 p^{8} T^{28} + 111552930858 p^{10} T^{30} + 1091292780 p^{12} T^{32} + 11956010 p^{14} T^{34} + 96305 p^{16} T^{36} + 462 p^{18} T^{38} + p^{20} T^{40} \) | |
| 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
−3.24641366397674209031039959668, −2.93909131322998702685632055413, −2.88436720418196435794431047553, −2.86239841259305768974580228491, −2.81505695504687666332580165212, −2.77522177467690277633888468602, −2.60705396125501939874202551622, −2.56118890833760863771194114275, −2.54699115423798817997084242166, −2.46250951536756980154792914610, −2.41900186342184431470972375026, −2.41760999189937237835821760019, −2.39465502709123643537571496622, −2.10915673935487434084590221172, −1.97947163791037452226236549593, −1.97211884931515380390769257810, −1.79934599351601310177150666027, −1.77895744205964595666200183357, −1.72467669670311623532054966464, −1.43101387033052295601617334744, −1.28062591285672985550731613511, −1.09652002898396276908543750617, −0.63770172990894976636477563180, −0.42269823178902754386477212446, −0.03197929863616733925232349133, 0.03197929863616733925232349133, 0.42269823178902754386477212446, 0.63770172990894976636477563180, 1.09652002898396276908543750617, 1.28062591285672985550731613511, 1.43101387033052295601617334744, 1.72467669670311623532054966464, 1.77895744205964595666200183357, 1.79934599351601310177150666027, 1.97211884931515380390769257810, 1.97947163791037452226236549593, 2.10915673935487434084590221172, 2.39465502709123643537571496622, 2.41760999189937237835821760019, 2.41900186342184431470972375026, 2.46250951536756980154792914610, 2.54699115423798817997084242166, 2.56118890833760863771194114275, 2.60705396125501939874202551622, 2.77522177467690277633888468602, 2.81505695504687666332580165212, 2.86239841259305768974580228491, 2.88436720418196435794431047553, 2.93909131322998702685632055413, 3.24641366397674209031039959668
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.