Dirichlet series
| L(s) = 1 | + 8·5-s − 8·9-s − 8·13-s + 16·25-s − 24·29-s + 40·37-s − 16·41-s − 64·45-s − 56·53-s + 8·61-s − 64·65-s + 32·73-s + 32·81-s + 32·89-s − 16·97-s + 40·101-s + 8·109-s + 64·117-s + 8·121-s − 56·125-s + 127-s + 131-s + 137-s + 139-s − 192·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 8/3·9-s − 2.21·13-s + 16/5·25-s − 4.45·29-s + 6.57·37-s − 2.49·41-s − 9.54·45-s − 7.69·53-s + 1.02·61-s − 7.93·65-s + 3.74·73-s + 32/9·81-s + 3.39·89-s − 1.62·97-s + 3.98·101-s + 0.766·109-s + 5.91·117-s + 8/11·121-s − 5.00·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.9·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{160}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{160}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{160}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.99239\times 10^{14}\) |
| Root analytic conductor: | \(2.85948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{160} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.822988341\) |
| \(L(\frac12)\) | \(\approx\) | \(2.822988341\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 8 T^{2} + 32 T^{4} + 152 T^{6} + 604 T^{8} + 1672 T^{10} + 5600 T^{12} + 16984 T^{14} + 42694 T^{16} + 16984 p^{2} T^{18} + 5600 p^{4} T^{20} + 1672 p^{6} T^{22} + 604 p^{8} T^{24} + 152 p^{10} T^{26} + 32 p^{12} T^{28} + 8 p^{14} T^{30} + p^{16} T^{32} \) |
| 5 | \( ( 1 - 4 T + 16 T^{2} - 36 T^{3} + 16 p T^{4} - 28 T^{5} - 144 T^{6} + 1156 T^{7} - 2546 T^{8} + 1156 p T^{9} - 144 p^{2} T^{10} - 28 p^{3} T^{11} + 16 p^{5} T^{12} - 36 p^{5} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 - 24 T^{4} - 2564 T^{8} + 76248 T^{12} + 4414470 T^{16} + 76248 p^{4} T^{20} - 2564 p^{8} T^{24} - 24 p^{12} T^{28} + p^{16} T^{32} \) | |
| 11 | \( 1 - 8 T^{2} + 32 T^{4} - 3032 T^{6} + 16732 T^{8} + 287416 T^{10} + 14560 p^{2} T^{12} + 280824 p T^{14} - 822624186 T^{16} + 280824 p^{3} T^{18} + 14560 p^{6} T^{20} + 287416 p^{6} T^{22} + 16732 p^{8} T^{24} - 3032 p^{10} T^{26} + 32 p^{12} T^{28} - 8 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 4 T + 16 T^{2} + 148 T^{3} + 528 T^{4} + 2412 T^{5} + 11952 T^{6} + 41148 T^{7} + 150862 T^{8} + 41148 p T^{9} + 11952 p^{2} T^{10} + 2412 p^{3} T^{11} + 528 p^{4} T^{12} + 148 p^{5} T^{13} + 16 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 64 T^{2} + 2412 T^{4} - 61760 T^{6} + 1207526 T^{8} - 61760 p^{2} T^{10} + 2412 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 24 T^{2} + 288 T^{4} - 20040 T^{6} + 447196 T^{8} - 6866712 T^{10} + 236809440 T^{12} - 4556359944 T^{14} + 61111615686 T^{16} - 4556359944 p^{2} T^{18} + 236809440 p^{4} T^{20} - 6866712 p^{6} T^{22} + 447196 p^{8} T^{24} - 20040 p^{10} T^{26} + 288 p^{12} T^{28} - 24 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 1368 T^{4} + 659068 T^{8} - 135138024 T^{12} + 27188126214 T^{16} - 135138024 p^{4} T^{20} + 659068 p^{8} T^{24} - 1368 p^{12} T^{28} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 12 T + 16 T^{2} - 244 T^{3} + 80 T^{4} + 7124 T^{5} - 14096 T^{6} - 276396 T^{7} - 1317042 T^{8} - 276396 p T^{9} - 14096 p^{2} T^{10} + 7124 p^{3} T^{11} + 80 p^{4} T^{12} - 244 p^{5} T^{13} + 16 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 24 T^{2} + 924 T^{4} + 32424 T^{6} - 181690 T^{8} + 32424 p^{2} T^{10} + 924 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 20 T + 160 T^{2} - 532 T^{3} - 560 T^{4} + 9140 T^{5} - 34720 T^{6} + 501556 T^{7} - 4885746 T^{8} + 501556 p T^{9} - 34720 p^{2} T^{10} + 9140 p^{3} T^{11} - 560 p^{4} T^{12} - 532 p^{5} T^{13} + 160 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 8 T + 32 T^{2} + 184 T^{3} + 1628 T^{4} + 10216 T^{5} + 46560 T^{6} + 75224 T^{7} - 1980282 T^{8} + 75224 p T^{9} + 46560 p^{2} T^{10} + 10216 p^{3} T^{11} + 1628 p^{4} T^{12} + 184 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 280 T^{2} + 39200 T^{4} - 3734920 T^{6} + 275251804 T^{8} - 16805127640 T^{10} + 890378725600 T^{12} - 42533679223560 T^{14} + 1886841963508806 T^{16} - 42533679223560 p^{2} T^{18} + 890378725600 p^{4} T^{20} - 16805127640 p^{6} T^{22} + 275251804 p^{8} T^{24} - 3734920 p^{10} T^{26} + 39200 p^{12} T^{28} - 280 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 120 T^{2} + 7644 T^{4} - 420168 T^{6} + 22246982 T^{8} - 420168 p^{2} T^{10} + 7644 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 28 T + 432 T^{2} + 5180 T^{3} + 53328 T^{4} + 486180 T^{5} + 4045072 T^{6} + 31068356 T^{7} + 227690766 T^{8} + 31068356 p T^{9} + 4045072 p^{2} T^{10} + 486180 p^{3} T^{11} + 53328 p^{4} T^{12} + 5180 p^{5} T^{13} + 432 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 216 T^{2} + 23328 T^{4} + 1766664 T^{6} + 107825500 T^{8} + 92970504 p T^{10} + 230013683424 T^{12} + 6568095595464 T^{14} + 171368345746758 T^{16} + 6568095595464 p^{2} T^{18} + 230013683424 p^{4} T^{20} + 92970504 p^{7} T^{22} + 107825500 p^{8} T^{24} + 1766664 p^{10} T^{26} + 23328 p^{12} T^{28} + 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 4 T + 144 T^{2} - 516 T^{3} + 10192 T^{4} - 35516 T^{5} + 220464 T^{6} - 1481724 T^{7} + 280782 T^{8} - 1481724 p T^{9} + 220464 p^{2} T^{10} - 35516 p^{3} T^{11} + 10192 p^{4} T^{12} - 516 p^{5} T^{13} + 144 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 24 T^{2} + 288 T^{4} - 196536 T^{6} - 46292900 T^{8} - 466437096 T^{10} + 21451064544 T^{12} + 4119669415944 T^{14} + 1054597662328518 T^{16} + 4119669415944 p^{2} T^{18} + 21451064544 p^{4} T^{20} - 466437096 p^{6} T^{22} - 46292900 p^{8} T^{24} - 196536 p^{10} T^{26} + 288 p^{12} T^{28} + 24 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 + 6952 T^{4} + 30802044 T^{8} - 214228188520 T^{12} - 1434458600506618 T^{16} - 214228188520 p^{4} T^{20} + 30802044 p^{8} T^{24} + 6952 p^{12} T^{28} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 10972 T^{4} - 51664 T^{5} + 221056 T^{6} + 1001424 T^{7} - 36752442 T^{8} + 1001424 p T^{9} + 221056 p^{2} T^{10} - 51664 p^{3} T^{11} + 10972 p^{4} T^{12} - 1264 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 504 T^{2} + 119004 T^{4} - 17120712 T^{6} + 1638907718 T^{8} - 17120712 p^{2} T^{10} + 119004 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 168 T^{2} + 14112 T^{4} - 1906488 T^{6} + 181015900 T^{8} - 9600901608 T^{10} + 875803336416 T^{12} - 61358369464248 T^{14} + 2408049709874118 T^{16} - 61358369464248 p^{2} T^{18} + 875803336416 p^{4} T^{20} - 9600901608 p^{6} T^{22} + 181015900 p^{8} T^{24} - 1906488 p^{10} T^{26} + 14112 p^{12} T^{28} - 168 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 16 T + 128 T^{2} - 1616 T^{3} + 28316 T^{4} - 281264 T^{5} + 2181504 T^{6} - 28123120 T^{7} + 359358150 T^{8} - 28123120 p T^{9} + 2181504 p^{2} T^{10} - 281264 p^{3} T^{11} + 28316 p^{4} T^{12} - 1616 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 4 T + 240 T^{2} + 860 T^{3} + 28118 T^{4} + 860 p T^{5} + 240 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−2.52654053749706861115801020778, −2.45539010972279177873502913124, −2.44545344294708756268026450262, −2.38395827150623597082700003431, −2.31018143074884540427580536328, −2.24127609871171617649213189049, −2.14714441797651357637774899999, −1.91242838900410131360155017352, −1.89212197947029395062451353314, −1.81123855517788203104141093057, −1.77284393665414699230227106879, −1.75728512786060875524401458811, −1.74329708134306540412101539375, −1.57230390196739897434792699460, −1.51907059868193256611227755134, −1.48464129980646797273981975259, −1.23189174058111699002930815077, −1.09002967874563692015262484591, −1.03244855982585706100797317445, −0.76868992471228077385462597536, −0.59950732465155207165700990487, −0.52638562732157862426549788648, −0.49881228538480653106413510910, −0.22726660910837873341283456252, −0.12677025860278484061920272270, 0.12677025860278484061920272270, 0.22726660910837873341283456252, 0.49881228538480653106413510910, 0.52638562732157862426549788648, 0.59950732465155207165700990487, 0.76868992471228077385462597536, 1.03244855982585706100797317445, 1.09002967874563692015262484591, 1.23189174058111699002930815077, 1.48464129980646797273981975259, 1.51907059868193256611227755134, 1.57230390196739897434792699460, 1.74329708134306540412101539375, 1.75728512786060875524401458811, 1.77284393665414699230227106879, 1.81123855517788203104141093057, 1.89212197947029395062451353314, 1.91242838900410131360155017352, 2.14714441797651357637774899999, 2.24127609871171617649213189049, 2.31018143074884540427580536328, 2.38395827150623597082700003431, 2.44545344294708756268026450262, 2.45539010972279177873502913124, 2.52654053749706861115801020778
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.