Dirichlet series
| L(s) = 1 | + 7-s + 6·11-s + 6·23-s + 16·25-s + 12·29-s + 4·37-s − 4·43-s − 2·49-s − 14·67-s + 6·77-s − 20·79-s − 20·109-s − 90·113-s − 25·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 6·161-s + 163-s + 167-s − 56·169-s + 173-s + 16·175-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.80·11-s + 1.25·23-s + 16/5·25-s + 2.22·29-s + 0.657·37-s − 0.609·43-s − 2/7·49-s − 1.71·67-s + 0.683·77-s − 2.25·79-s − 1.91·109-s − 8.46·113-s − 2.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.472·161-s + 0.0783·163-s + 0.0773·167-s − 4.30·169-s + 0.0760·173-s + 1.20·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{48} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.33580\times 10^{22}\) |
| Root analytic conductor: | \(4.91393\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(33.30957974\) |
| \(L(\frac12)\) | \(\approx\) | \(33.30957974\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - T + 3 T^{2} - 20 T^{3} + 53 T^{4} - 3 p T^{5} - 74 T^{6} - 355 T^{7} - 1314 T^{8} - 355 p T^{9} - 74 p^{2} T^{10} - 3 p^{4} T^{11} + 53 p^{4} T^{12} - 20 p^{5} T^{13} + 3 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 16 T^{2} + 159 T^{4} - 998 T^{6} + 4298 T^{8} - 12066 T^{10} + 21316 T^{12} - 64633 T^{14} + 271566 T^{16} - 64633 p^{2} T^{18} + 21316 p^{4} T^{20} - 12066 p^{6} T^{22} + 4298 p^{8} T^{24} - 998 p^{10} T^{26} + 159 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 3 T + 26 T^{2} - 69 T^{3} + 265 T^{4} - 756 T^{5} + 235 p T^{6} - 8259 T^{7} + 34846 T^{8} - 8259 p T^{9} + 235 p^{3} T^{10} - 756 p^{3} T^{11} + 265 p^{4} T^{12} - 69 p^{5} T^{13} + 26 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 363405056 p^{2} T^{18} + 12863830 p^{4} T^{20} + 1615656 p^{6} T^{22} + 274865 p^{8} T^{24} + 26848 p^{10} T^{26} + 1590 p^{12} T^{28} + 56 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 58 T^{2} + 1603 T^{4} + 30013 T^{6} + 497674 T^{8} + 30013 p^{2} T^{10} + 1603 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 77 T^{2} + 3034 T^{4} - 84368 T^{6} + 1811980 T^{8} - 84368 p^{2} T^{10} + 3034 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 3 T + 56 T^{2} - 159 T^{3} + 1321 T^{4} - 3510 T^{5} + 30863 T^{6} - 67173 T^{7} + 842170 T^{8} - 67173 p T^{9} + 30863 p^{2} T^{10} - 3510 p^{3} T^{11} + 1321 p^{4} T^{12} - 159 p^{5} T^{13} + 56 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 6 T + 53 T^{2} - 246 T^{3} + 20 p T^{4} - 540 T^{5} - 22066 T^{6} + 299625 T^{7} - 1268282 T^{8} + 299625 p T^{9} - 22066 p^{2} T^{10} - 540 p^{3} T^{11} + 20 p^{5} T^{12} - 246 p^{5} T^{13} + 53 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( 1 + 176 T^{2} + 16407 T^{4} + 1045570 T^{6} + 50906954 T^{8} + 2036295270 T^{10} + 71233973380 T^{12} + 2306080860179 T^{14} + 72156636517806 T^{16} + 2306080860179 p^{2} T^{18} + 71233973380 p^{4} T^{20} + 2036295270 p^{6} T^{22} + 50906954 p^{8} T^{24} + 1045570 p^{10} T^{26} + 16407 p^{12} T^{28} + 176 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - T + 82 T^{2} - 88 T^{3} + 3940 T^{4} - 88 p T^{5} + 82 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 41 | \( 1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 1573469011633 p^{2} T^{18} + 30503470855 p^{4} T^{20} - 588962616 p^{6} T^{22} + 12352577 p^{8} T^{24} - 353897 p^{10} T^{26} + 9564 p^{12} T^{28} - 151 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 2 T - 93 T^{2} + 64 T^{3} + 92 p T^{4} - 12192 T^{5} - 158990 T^{6} + 321485 T^{7} + 7972668 T^{8} + 321485 p T^{9} - 158990 p^{2} T^{10} - 12192 p^{3} T^{11} + 92 p^{5} T^{12} + 64 p^{5} T^{13} - 93 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 154 T^{2} + 7683 T^{4} - 245888 T^{6} + 25028264 T^{8} - 1538143278 T^{10} + 38475997042 T^{12} - 55371690107 p T^{14} + 205422570296046 T^{16} - 55371690107 p^{3} T^{18} + 38475997042 p^{4} T^{20} - 1538143278 p^{6} T^{22} + 25028264 p^{8} T^{24} - 245888 p^{10} T^{26} + 7683 p^{12} T^{28} - 154 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 10 T^{2} + 3133 T^{4} + 54785 T^{6} + 5778574 T^{8} + 54785 p^{2} T^{10} + 3133 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 376 T^{2} + 75453 T^{4} - 10695122 T^{6} + 1193881406 T^{8} - 111003832932 T^{10} + 8882998756942 T^{12} - 624280670070511 T^{14} + 38988958528434078 T^{16} - 624280670070511 p^{2} T^{18} + 8882998756942 p^{4} T^{20} - 111003832932 p^{6} T^{22} + 1193881406 p^{8} T^{24} - 10695122 p^{10} T^{26} + 75453 p^{12} T^{28} - 376 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 137 T^{2} + 306 T^{4} - 368483 T^{6} + 33183791 T^{8} + 2225199582 T^{10} - 143899132121 T^{12} + 1205057298539 T^{14} + 1133086682080440 T^{16} + 1205057298539 p^{2} T^{18} - 143899132121 p^{4} T^{20} + 2225199582 p^{6} T^{22} + 33183791 p^{8} T^{24} - 368483 p^{10} T^{26} + 306 p^{12} T^{28} + 137 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 7 T - 108 T^{2} + 293 T^{3} + 11753 T^{4} - 50778 T^{5} - 198827 T^{6} + 2620663 T^{7} - 10701486 T^{8} + 2620663 p T^{9} - 198827 p^{2} T^{10} - 50778 p^{3} T^{11} + 11753 p^{4} T^{12} + 293 p^{5} T^{13} - 108 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 7488268 p^{2} T^{10} + 64216 p^{4} T^{12} - 361 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 341 T^{2} + 62398 T^{4} - 7562288 T^{6} + 647645452 T^{8} - 7562288 p^{2} T^{10} + 62398 p^{4} T^{12} - 341 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 10 T - 123 T^{2} - 844 T^{3} + 11042 T^{4} - 2166 T^{5} - 1268066 T^{6} + 531391 T^{7} + 106354368 T^{8} + 531391 p T^{9} - 1268066 p^{2} T^{10} - 2166 p^{3} T^{11} + 11042 p^{4} T^{12} - 844 p^{5} T^{13} - 123 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 849901181197043 p^{2} T^{18} - 2507933243939 p^{4} T^{20} - 57837981066 p^{6} T^{22} + 1185510017 p^{8} T^{24} - 12501089 p^{10} T^{26} + 87990 p^{12} T^{28} - 397 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 64 T^{2} + 9487 T^{4} + 662335 T^{6} + 143358802 T^{8} + 662335 p^{2} T^{10} + 9487 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 591335472 p T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 785812116013063 p^{2} T^{18} - 1663618077194 p^{4} T^{20} + 591335472 p^{7} T^{22} + 1046175986 p^{8} T^{24} + 10963666 p^{10} T^{26} + 81609 p^{12} T^{28} + 404 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.07596553879716633702976522798, −1.96804047256266640807142408239, −1.87933232143127857629306052779, −1.79405597033120230916445193132, −1.72632455119061607900387889351, −1.68577754863703117971981796876, −1.67704857283111193267310002229, −1.55236482163628234385710186263, −1.53666274506441651245836964092, −1.52030698287082932301813548108, −1.50302582506875788749245059018, −1.31187155333133311222775405115, −1.15531953264181335639913175370, −1.12716544531974737354904386755, −1.05748802766411680906821475772, −1.05467922869144439050684399058, −0.995157788707012540103276681691, −0.906437062611725769971823260413, −0.879258222239754047363451656087, −0.73188380009712035298654417598, −0.42916984662637562173461469756, −0.31962303695736565687435100641, −0.28791609949281964360567405605, −0.28288769921923298766090369043, −0.19089447436608539075278025437, 0.19089447436608539075278025437, 0.28288769921923298766090369043, 0.28791609949281964360567405605, 0.31962303695736565687435100641, 0.42916984662637562173461469756, 0.73188380009712035298654417598, 0.879258222239754047363451656087, 0.906437062611725769971823260413, 0.995157788707012540103276681691, 1.05467922869144439050684399058, 1.05748802766411680906821475772, 1.12716544531974737354904386755, 1.15531953264181335639913175370, 1.31187155333133311222775405115, 1.50302582506875788749245059018, 1.52030698287082932301813548108, 1.53666274506441651245836964092, 1.55236482163628234385710186263, 1.67704857283111193267310002229, 1.68577754863703117971981796876, 1.72632455119061607900387889351, 1.79405597033120230916445193132, 1.87933232143127857629306052779, 1.96804047256266640807142408239, 2.07596553879716633702976522798
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.