Dirichlet series
| L(s) = 1 | + 10·3-s + 5·4-s + 45·9-s + 50·12-s + 4·13-s + 10·16-s − 6·17-s − 16·23-s − 24·25-s + 110·27-s − 28·29-s + 225·36-s + 40·39-s + 24·43-s + 100·48-s + 49-s − 60·51-s + 20·52-s − 22·53-s + 14·61-s + 5·64-s − 30·68-s − 160·69-s − 240·75-s + 4·79-s + 110·81-s − 280·87-s + ⋯ |
| L(s) = 1 | + 5.77·3-s + 5/2·4-s + 15·9-s + 14.4·12-s + 1.10·13-s + 5/2·16-s − 1.45·17-s − 3.33·23-s − 4.79·25-s + 21.1·27-s − 5.19·29-s + 75/2·36-s + 6.40·39-s + 3.65·43-s + 14.4·48-s + 1/7·49-s − 8.40·51-s + 2.77·52-s − 3.02·53-s + 1.79·61-s + 5/8·64-s − 3.63·68-s − 19.2·69-s − 27.7·75-s + 0.450·79-s + 12.2·81-s − 30.0·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.15736\times 10^{12}\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 13^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.801940457\) |
| \(L(\frac12)\) | \(\approx\) | \(5.801940457\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{5} \) |
| 3 | \( ( 1 - T + T^{2} )^{10} \) | |
| 7 | \( 1 - T^{2} + 3 T^{4} - 50 p T^{6} + 701 T^{8} + 8745 T^{10} + 701 p^{2} T^{12} - 50 p^{5} T^{14} + 3 p^{6} T^{16} - p^{8} T^{18} + p^{10} T^{20} \) | |
| 13 | \( ( 1 - 2 T + 19 T^{2} - 4 p T^{3} - 16 T^{4} - 724 T^{5} - 16 p T^{6} - 4 p^{3} T^{7} + 19 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| good | 5 | \( 1 + 24 T^{2} + 246 T^{4} + 324 p T^{6} + 10531 T^{8} + 74628 T^{10} + 94844 p T^{12} + 113844 p^{2} T^{14} + 16600861 T^{16} + 16782552 p T^{18} + 16014178 p^{2} T^{20} + 16782552 p^{3} T^{22} + 16600861 p^{4} T^{24} + 113844 p^{8} T^{26} + 94844 p^{9} T^{28} + 74628 p^{10} T^{30} + 10531 p^{12} T^{32} + 324 p^{15} T^{34} + 246 p^{16} T^{36} + 24 p^{18} T^{38} + p^{20} T^{40} \) |
| 11 | \( 1 + 25 T^{2} + 26 p T^{4} - 3617 T^{6} - 121378 T^{8} - 1310865 T^{10} + 5869888 T^{12} + 252746167 T^{14} + 2159832905 T^{16} - 13902198830 T^{18} - 360621176460 T^{20} - 13902198830 p^{2} T^{22} + 2159832905 p^{4} T^{24} + 252746167 p^{6} T^{26} + 5869888 p^{8} T^{28} - 1310865 p^{10} T^{30} - 121378 p^{12} T^{32} - 3617 p^{14} T^{34} + 26 p^{17} T^{36} + 25 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( ( 1 + 3 T - 25 T^{2} + 124 T^{3} + 1062 T^{4} - 3190 T^{5} + 6283 T^{6} + 141573 T^{7} - 265987 T^{8} - 321446 T^{9} + 13097064 T^{10} - 321446 p T^{11} - 265987 p^{2} T^{12} + 141573 p^{3} T^{13} + 6283 p^{4} T^{14} - 3190 p^{5} T^{15} + 1062 p^{6} T^{16} + 124 p^{7} T^{17} - 25 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 19 | \( 1 + 123 T^{2} + 7993 T^{4} + 354190 T^{6} + 11757898 T^{8} + 300618558 T^{10} + 5814680167 T^{12} + 76088816781 T^{14} + 293567055183 T^{16} - 15388455318172 T^{18} - 456324954017300 T^{20} - 15388455318172 p^{2} T^{22} + 293567055183 p^{4} T^{24} + 76088816781 p^{6} T^{26} + 5814680167 p^{8} T^{28} + 300618558 p^{10} T^{30} + 11757898 p^{12} T^{32} + 354190 p^{14} T^{34} + 7993 p^{16} T^{36} + 123 p^{18} T^{38} + p^{20} T^{40} \) | |
| 23 | \( ( 1 + 8 T - 31 T^{2} - 244 T^{3} + 1207 T^{4} + 2458 T^{5} - 59094 T^{6} - 90270 T^{7} + 1409697 T^{8} + 1512890 T^{9} - 27151625 T^{10} + 1512890 p T^{11} + 1409697 p^{2} T^{12} - 90270 p^{3} T^{13} - 59094 p^{4} T^{14} + 2458 p^{5} T^{15} + 1207 p^{6} T^{16} - 244 p^{7} T^{17} - 31 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 7 T + 81 T^{2} + 404 T^{3} + 3409 T^{4} + 15627 T^{5} + 3409 p T^{6} + 404 p^{2} T^{7} + 81 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 31 | \( 1 + 228 T^{2} + 26866 T^{4} + 2211844 T^{6} + 143860267 T^{8} + 7883636520 T^{10} + 377713164472 T^{12} + 16167032122824 T^{14} + 626185477434285 T^{16} + 22118876665285292 T^{18} + 715801597978391938 T^{20} + 22118876665285292 p^{2} T^{22} + 626185477434285 p^{4} T^{24} + 16167032122824 p^{6} T^{26} + 377713164472 p^{8} T^{28} + 7883636520 p^{10} T^{30} + 143860267 p^{12} T^{32} + 2211844 p^{14} T^{34} + 26866 p^{16} T^{36} + 228 p^{18} T^{38} + p^{20} T^{40} \) | |
| 37 | \( 1 + 130 T^{2} + 8911 T^{4} + 395778 T^{6} + 12211455 T^{8} + 318109872 T^{10} + 319343370 p T^{12} + 613083351768 T^{14} + 26706587678781 T^{16} + 926429566048594 T^{18} + 31426625318592673 T^{20} + 926429566048594 p^{2} T^{22} + 26706587678781 p^{4} T^{24} + 613083351768 p^{6} T^{26} + 319343370 p^{9} T^{28} + 318109872 p^{10} T^{30} + 12211455 p^{12} T^{32} + 395778 p^{14} T^{34} + 8911 p^{16} T^{36} + 130 p^{18} T^{38} + p^{20} T^{40} \) | |
| 41 | \( ( 1 + 34 T^{2} + 7269 T^{4} + 236620 T^{6} + 22529050 T^{8} + 603259716 T^{10} + 22529050 p^{2} T^{12} + 236620 p^{4} T^{14} + 7269 p^{6} T^{16} + 34 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 6 T + 149 T^{2} - 796 T^{3} + 11088 T^{4} - 47980 T^{5} + 11088 p T^{6} - 796 p^{2} T^{7} + 149 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 47 | \( 1 + 271 T^{2} + 35881 T^{4} + 3126166 T^{6} + 210184286 T^{8} + 12747174834 T^{10} + 778029516067 T^{12} + 46628291795569 T^{14} + 2547460090955747 T^{16} + 125686487343189352 T^{18} + 5924933119352668164 T^{20} + 125686487343189352 p^{2} T^{22} + 2547460090955747 p^{4} T^{24} + 46628291795569 p^{6} T^{26} + 778029516067 p^{8} T^{28} + 12747174834 p^{10} T^{30} + 210184286 p^{12} T^{32} + 3126166 p^{14} T^{34} + 35881 p^{16} T^{36} + 271 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( ( 1 + 11 T - 112 T^{2} - 25 p T^{3} + 9432 T^{4} + 83341 T^{5} - 816680 T^{6} - 3629503 T^{7} + 61191143 T^{8} + 89102100 T^{9} - 3477029424 T^{10} + 89102100 p T^{11} + 61191143 p^{2} T^{12} - 3629503 p^{3} T^{13} - 816680 p^{4} T^{14} + 83341 p^{5} T^{15} + 9432 p^{6} T^{16} - 25 p^{8} T^{17} - 112 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 59 | \( 1 + 221 T^{2} + 20106 T^{4} + 710111 T^{6} - 20629214 T^{8} - 3046559021 T^{10} - 111402587480 T^{12} + 731502200803 T^{14} + 407320291351877 T^{16} + 42936030210930526 T^{18} + 3054910087623681484 T^{20} + 42936030210930526 p^{2} T^{22} + 407320291351877 p^{4} T^{24} + 731502200803 p^{6} T^{26} - 111402587480 p^{8} T^{28} - 3046559021 p^{10} T^{30} - 20629214 p^{12} T^{32} + 710111 p^{14} T^{34} + 20106 p^{16} T^{36} + 221 p^{18} T^{38} + p^{20} T^{40} \) | |
| 61 | \( ( 1 - 7 T - 117 T^{2} - 392 T^{3} + 15098 T^{4} + 74378 T^{5} - 270105 T^{6} - 9893333 T^{7} - 20548315 T^{8} + 166528230 T^{9} + 4623082064 T^{10} + 166528230 p T^{11} - 20548315 p^{2} T^{12} - 9893333 p^{3} T^{13} - 270105 p^{4} T^{14} + 74378 p^{5} T^{15} + 15098 p^{6} T^{16} - 392 p^{7} T^{17} - 117 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 + 207 T^{2} + 11641 T^{4} - 557914 T^{6} - 63577194 T^{8} + 3979670834 T^{10} + 574917939283 T^{12} - 1986356709687 T^{14} - 2046008523704909 T^{16} + 67260945400183048 T^{18} + 16669431511350954804 T^{20} + 67260945400183048 p^{2} T^{22} - 2046008523704909 p^{4} T^{24} - 1986356709687 p^{6} T^{26} + 574917939283 p^{8} T^{28} + 3979670834 p^{10} T^{30} - 63577194 p^{12} T^{32} - 557914 p^{14} T^{34} + 11641 p^{16} T^{36} + 207 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 - 251 T^{2} + 44136 T^{4} - 5424801 T^{6} + 535413747 T^{8} - 41868035424 T^{10} + 535413747 p^{2} T^{12} - 5424801 p^{4} T^{14} + 44136 p^{6} T^{16} - 251 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 + 558 T^{2} + 161299 T^{4} + 32905174 T^{6} + 5346992203 T^{8} + 731384567616 T^{10} + 86772131688406 T^{12} + 9105054667525488 T^{14} + 855370708345231125 T^{16} + 72446211808799395814 T^{18} + \)\(55\!\cdots\!93\)\( T^{20} + 72446211808799395814 p^{2} T^{22} + 855370708345231125 p^{4} T^{24} + 9105054667525488 p^{6} T^{26} + 86772131688406 p^{8} T^{28} + 731384567616 p^{10} T^{30} + 5346992203 p^{12} T^{32} + 32905174 p^{14} T^{34} + 161299 p^{16} T^{36} + 558 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 2 T - 232 T^{2} - 1520 T^{3} + 33777 T^{4} + 321512 T^{5} - 1304890 T^{6} - 38346354 T^{7} - 85685519 T^{8} + 1227431736 T^{9} + 19534711118 T^{10} + 1227431736 p T^{11} - 85685519 p^{2} T^{12} - 38346354 p^{3} T^{13} - 1304890 p^{4} T^{14} + 321512 p^{5} T^{15} + 33777 p^{6} T^{16} - 1520 p^{7} T^{17} - 232 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 - 348 T^{2} + 55046 T^{4} - 5238910 T^{6} + 338142121 T^{8} - 21740336636 T^{10} + 338142121 p^{2} T^{12} - 5238910 p^{4} T^{14} + 55046 p^{6} T^{16} - 348 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 89 | \( 1 + 318 T^{2} + 32899 T^{4} + 1025542 T^{6} + 97229643 T^{8} + 20693457424 T^{10} + 847466504374 T^{12} + 81746078346384 T^{14} + 26626162382705893 T^{16} + 1869157707738386294 T^{18} + 46637029842007593705 T^{20} + 1869157707738386294 p^{2} T^{22} + 26626162382705893 p^{4} T^{24} + 81746078346384 p^{6} T^{26} + 847466504374 p^{8} T^{28} + 20693457424 p^{10} T^{30} + 97229643 p^{12} T^{32} + 1025542 p^{14} T^{34} + 32899 p^{16} T^{36} + 318 p^{18} T^{38} + p^{20} T^{40} \) | |
| 97 | \( ( 1 - 76 T^{2} + 20730 T^{4} - 2758262 T^{6} + 302277473 T^{8} - 33146313744 T^{10} + 302277473 p^{2} T^{12} - 2758262 p^{4} T^{14} + 20730 p^{6} T^{16} - 76 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 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
−2.40572655532119693815331961337, −2.39676978301935963765291517638, −2.36718548047617975490725914280, −2.36557320225474341248318481163, −2.34480543328714188314843478289, −2.29359339384885112555973728525, −2.29115694337687445486355840904, −2.07271458455986985916146775927, −2.00849937581288247519591392346, −1.91531587533299051372696520132, −1.80053674958784570792516127507, −1.73542502308649536687931666438, −1.72412695782793516730037997044, −1.56315060232583956956166894545, −1.54738700758617357011416827078, −1.54069983861164246958103338623, −1.47946837832586067353110970447, −1.31977872919570090902761125662, −1.30909471666910613291932924508, −1.19107364346317785511268681745, −0.75063407308225408037255608966, −0.69276095247967463489072628279, −0.38611898093966015570484685519, −0.19483960100297875299586249542, −0.098622291248667760493202697339, 0.098622291248667760493202697339, 0.19483960100297875299586249542, 0.38611898093966015570484685519, 0.69276095247967463489072628279, 0.75063407308225408037255608966, 1.19107364346317785511268681745, 1.30909471666910613291932924508, 1.31977872919570090902761125662, 1.47946837832586067353110970447, 1.54069983861164246958103338623, 1.54738700758617357011416827078, 1.56315060232583956956166894545, 1.72412695782793516730037997044, 1.73542502308649536687931666438, 1.80053674958784570792516127507, 1.91531587533299051372696520132, 2.00849937581288247519591392346, 2.07271458455986985916146775927, 2.29115694337687445486355840904, 2.29359339384885112555973728525, 2.34480543328714188314843478289, 2.36557320225474341248318481163, 2.36718548047617975490725914280, 2.39676978301935963765291517638, 2.40572655532119693815331961337
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.