Dirichlet series
| L(s) = 1 | + 8·5-s + 8·9-s + 24·13-s + 48·25-s − 24·29-s + 40·37-s + 16·41-s + 64·45-s − 56·53-s + 8·61-s + 192·65-s + 32·73-s + 32·81-s − 32·89-s + 16·97-s + 40·101-s + 8·109-s + 192·117-s − 8·121-s + 232·125-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 + 6.65·13-s + 48/5·25-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 + 23.8·65-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 + 17.7·117-s − 0.727·121-s + 20.7·125-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
| \(L(1)\) | \(\approx\) | \(360.5021001\) |
| \(L(\frac12)\) | \(\approx\) | \(360.5021001\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 8 T^{2} + 32 T^{4} - 88 T^{6} + 92 T^{8} + 184 T^{10} - 544 T^{12} - 472 T^{14} + 5830 T^{16} - 472 p^{2} T^{18} - 544 p^{4} T^{20} + 184 p^{6} T^{22} + 92 p^{8} T^{24} - 88 p^{10} T^{26} + 32 p^{12} T^{28} - 8 p^{14} T^{30} + p^{16} T^{32} \) |
| 5 | \( ( 1 - 4 T + 12 T^{3} + 16 T^{4} - 12 T^{5} - 192 T^{6} - 188 T^{7} + 1934 T^{8} - 188 p T^{9} - 192 p^{2} T^{10} - 12 p^{3} T^{11} + 16 p^{4} T^{12} + 12 p^{5} T^{13} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 - 24 T^{4} + 6652 T^{8} - 108072 T^{12} + 428262 p^{2} T^{16} - 108072 p^{4} T^{20} + 6652 p^{8} T^{24} - 24 p^{12} T^{28} + p^{16} T^{32} \) | |
| 11 | \( 1 + 8 T^{2} + 32 T^{4} + 1048 T^{6} + 31068 T^{8} + 197768 T^{10} + 1137120 T^{12} + 26161624 T^{14} + 635040838 T^{16} + 26161624 p^{2} T^{18} + 1137120 p^{4} T^{20} + 197768 p^{6} T^{22} + 31068 p^{8} T^{24} + 1048 p^{10} T^{26} + 32 p^{12} T^{28} + 8 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 - 12 T + 80 T^{2} - 396 T^{3} + 1616 T^{4} - 6516 T^{5} + 28272 T^{6} - 122676 T^{7} + 478286 T^{8} - 122676 p T^{9} + 28272 p^{2} T^{10} - 6516 p^{3} T^{11} + 1616 p^{4} T^{12} - 396 p^{5} T^{13} + 80 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 64 T^{2} + 2156 T^{4} - 53568 T^{6} + 1042150 T^{8} - 53568 p^{2} T^{10} + 2156 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 + 24 T^{2} + 288 T^{4} + 5640 T^{6} + 41692 T^{8} - 306216 T^{10} - 3451680 T^{12} - 430031736 T^{14} - 19133875002 T^{16} - 430031736 p^{2} T^{18} - 3451680 p^{4} T^{20} - 306216 p^{6} T^{22} + 41692 p^{8} T^{24} + 5640 p^{10} T^{26} + 288 p^{12} T^{28} + 24 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 344 T^{4} - 485764 T^{8} + 91787544 T^{12} + 141496848902 T^{16} + 91787544 p^{4} T^{20} - 485764 p^{8} T^{24} - 344 p^{12} T^{28} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 12 T + 32 T^{2} - 196 T^{3} - 1264 T^{4} + 2404 T^{5} + 53792 T^{6} + 107796 T^{7} - 559666 T^{8} + 107796 p T^{9} + 53792 p^{2} T^{10} + 2404 p^{3} T^{11} - 1264 p^{4} T^{12} - 196 p^{5} T^{13} + 32 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 108 T^{2} + 4806 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{4} \) | |
| 37 | \( ( 1 - 20 T + 128 T^{2} - 548 T^{3} + 7952 T^{4} - 80188 T^{5} + 454400 T^{6} - 2684236 T^{7} + 18087182 T^{8} - 2684236 p T^{9} + 454400 p^{2} T^{10} - 80188 p^{3} T^{11} + 7952 p^{4} T^{12} - 548 p^{5} T^{13} + 128 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 8 T + 32 T^{2} + 264 T^{3} - 548 T^{4} - 3112 T^{5} + 77280 T^{6} + 489128 T^{7} - 3563130 T^{8} + 489128 p T^{9} + 77280 p^{2} T^{10} - 3112 p^{3} T^{11} - 548 p^{4} T^{12} + 264 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 88 T^{2} + 3872 T^{4} - 16888 T^{6} - 10463652 T^{8} - 584492840 T^{10} - 10777507104 T^{12} + 668880324680 T^{14} + 49697523959878 T^{16} + 668880324680 p^{2} T^{18} - 10777507104 p^{4} T^{20} - 584492840 p^{6} T^{22} - 10463652 p^{8} T^{24} - 16888 p^{10} T^{26} + 3872 p^{12} T^{28} + 88 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 248 T^{2} + 29148 T^{4} - 2187464 T^{6} + 118603334 T^{8} - 2187464 p^{2} T^{10} + 29148 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 28 T + 304 T^{2} + 364 T^{3} - 28272 T^{4} - 331660 T^{5} - 1040240 T^{6} + 12818372 T^{7} + 170431502 T^{8} + 12818372 p T^{9} - 1040240 p^{2} T^{10} - 331660 p^{3} T^{11} - 28272 p^{4} T^{12} + 364 p^{5} T^{13} + 304 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 - 24 T^{2} + 288 T^{4} + 258168 T^{6} - 21141668 T^{8} + 593418024 T^{10} + 25172125920 T^{12} - 1782907948296 T^{14} + 319079688144198 T^{16} - 1782907948296 p^{2} T^{18} + 25172125920 p^{4} T^{20} + 593418024 p^{6} T^{22} - 21141668 p^{8} T^{24} + 258168 p^{10} T^{26} + 288 p^{12} T^{28} - 24 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 4 T - 144 T^{2} + 1196 T^{3} + 7952 T^{4} - 100460 T^{5} - 95024 T^{6} + 2674948 T^{7} - 8271154 T^{8} + 2674948 p T^{9} - 95024 p^{2} T^{10} - 100460 p^{3} T^{11} + 7952 p^{4} T^{12} + 1196 p^{5} T^{13} - 144 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 216 T^{2} + 23328 T^{4} - 1657800 T^{6} + 78886492 T^{8} - 82811160 T^{10} - 448226454816 T^{12} + 49748087534904 T^{14} - 3700043386949946 T^{16} + 49748087534904 p^{2} T^{18} - 448226454816 p^{4} T^{20} - 82811160 p^{6} T^{22} + 78886492 p^{8} T^{24} - 1657800 p^{10} T^{26} + 23328 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 24792 T^{4} + 325990524 T^{8} - 2768742343016 T^{12} + 16536352607226630 T^{16} - 2768742343016 p^{4} T^{20} + 325990524 p^{8} T^{24} - 24792 p^{12} T^{28} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 16 T + 128 T^{2} - 1584 T^{3} + 24540 T^{4} - 206352 T^{5} + 1415040 T^{6} - 15474352 T^{7} + 168829382 T^{8} - 15474352 p T^{9} + 1415040 p^{2} T^{10} - 206352 p^{3} T^{11} + 24540 p^{4} T^{12} - 1584 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 120 T^{2} + 15324 T^{4} - 1678152 T^{6} + 117571910 T^{8} - 1678152 p^{2} T^{10} + 15324 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 216 T^{2} + 23328 T^{4} - 10584 p T^{6} + 13673308 T^{8} - 4254474456 T^{10} + 985852080864 T^{12} - 64310503083336 T^{14} + 4328449978430406 T^{16} - 64310503083336 p^{2} T^{18} + 985852080864 p^{4} T^{20} - 4254474456 p^{6} T^{22} + 13673308 p^{8} T^{24} - 10584 p^{11} T^{26} + 23328 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 16 T + 128 T^{2} + 1104 T^{3} + 1948 T^{4} - 15184 T^{5} + 117120 T^{6} + 3790832 T^{7} + 66739398 T^{8} + 3790832 p T^{9} + 117120 p^{2} T^{10} - 15184 p^{3} T^{11} + 1948 p^{4} T^{12} + 1104 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 4 T + 352 T^{2} - 1180 T^{3} + 49526 T^{4} - 1180 p T^{5} + 352 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−2.53600268740320581879058014472, −2.48791210522614570283918441271, −2.35269146799655787619811597282, −2.32122337352128585915733188103, −2.17362011021960724982795477562, −2.10387504127283683561812703788, −2.09354339487738476623999640649, −2.04409572115960981646456515340, −1.94652868203560751569436111991, −1.83875741640425135004378891429, −1.68980808901866865006814343539, −1.67844176190880149081289651801, −1.37075379152253187755445832267, −1.36705537474542317708740609145, −1.31105966343929282618981484521, −1.27435942290469217947428056592, −1.25717925560309613086482086409, −1.19263057235737229629694382045, −1.17173877854830280520946771290, −1.14640411905707672814517875215, −0.865459075323616257462728658211, −0.77775498621428568589929139786, −0.51483661865057620604203197377, −0.48377395331635139745776454981, −0.32540050011981413906178336287, 0.32540050011981413906178336287, 0.48377395331635139745776454981, 0.51483661865057620604203197377, 0.77775498621428568589929139786, 0.865459075323616257462728658211, 1.14640411905707672814517875215, 1.17173877854830280520946771290, 1.19263057235737229629694382045, 1.25717925560309613086482086409, 1.27435942290469217947428056592, 1.31105966343929282618981484521, 1.36705537474542317708740609145, 1.37075379152253187755445832267, 1.67844176190880149081289651801, 1.68980808901866865006814343539, 1.83875741640425135004378891429, 1.94652868203560751569436111991, 2.04409572115960981646456515340, 2.09354339487738476623999640649, 2.10387504127283683561812703788, 2.17362011021960724982795477562, 2.32122337352128585915733188103, 2.35269146799655787619811597282, 2.48791210522614570283918441271, 2.53600268740320581879058014472
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.