Dirichlet series
| L(s) = 1 | − 2-s + 6·7-s + 8-s − 6·14-s − 3·16-s + 28·17-s + 10·23-s − 19·25-s − 10·31-s − 28·34-s + 8·41-s − 10·46-s − 6·47-s + 55·49-s + 19·50-s + 6·56-s + 10·62-s + 7·64-s − 72·71-s − 44·73-s − 30·79-s − 8·82-s − 64·89-s + 6·94-s − 55·98-s − 14·103-s − 18·112-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.26·7-s + 0.353·8-s − 1.60·14-s − 3/4·16-s + 6.79·17-s + 2.08·23-s − 3.79·25-s − 1.79·31-s − 4.80·34-s + 1.24·41-s − 1.47·46-s − 0.875·47-s + 55/7·49-s + 2.68·50-s + 0.801·56-s + 1.27·62-s + 7/8·64-s − 8.54·71-s − 5.14·73-s − 3.37·79-s − 0.883·82-s − 6.78·89-s + 0.618·94-s − 5.55·98-s − 1.37·103-s − 1.70·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6133.29\) |
| Root analytic conductor: | \(1.31330\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{48} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.332267525\) |
| \(L(\frac12)\) | \(\approx\) | \(1.332267525\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + T + T^{2} + p T^{4} + p^{2} T^{5} + p^{3} T^{7} + p^{2} T^{8} + p^{4} T^{9} + p^{5} T^{11} + p^{5} T^{12} + p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 631122 p^{2} T^{18} - 61926 p^{4} T^{20} + 4928 p^{6} T^{22} + 3893 p^{8} T^{24} + 1031 p^{10} T^{26} + 176 p^{12} T^{28} + 19 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( ( 1 - 3 T - 2 p T^{2} + 39 T^{3} + 139 T^{4} - 36 p T^{5} - 1208 T^{6} + 666 T^{7} + 9424 T^{8} + 666 p T^{9} - 1208 p^{2} T^{10} - 36 p^{4} T^{11} + 139 p^{4} T^{12} + 39 p^{5} T^{13} - 2 p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 11 | \( 1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 158095260 p^{2} T^{18} + 21031138 p^{4} T^{20} + 2632140 p^{6} T^{22} + 242041 p^{8} T^{24} + 17592 p^{10} T^{26} + 1090 p^{12} T^{28} + 48 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 5665 p T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 411982806 p^{2} T^{18} - 5456606 p^{4} T^{20} - 1059264 p^{6} T^{22} + 5665 p^{9} T^{24} + 16911 p^{10} T^{26} + 1288 p^{12} T^{28} + 51 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 7 T + 66 T^{2} - 309 T^{3} + 1702 T^{4} - 309 p T^{5} + 66 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 19 | \( ( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 53763 p^{2} T^{10} + 2306 p^{4} T^{12} - 69 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 5 T - 32 T^{2} - 45 T^{3} + 1385 T^{4} + 3040 T^{5} - 11142 T^{6} - 48640 T^{7} - 123716 T^{8} - 48640 p T^{9} - 11142 p^{2} T^{10} + 3040 p^{3} T^{11} + 1385 p^{4} T^{12} - 45 p^{5} T^{13} - 32 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 328820871018 p^{2} T^{18} - 3705324014 p^{4} T^{20} + 144056448 p^{6} T^{22} + 10067965 p^{8} T^{24} + 340647 p^{10} T^{26} + 7912 p^{12} T^{28} + 123 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 5 T - 48 T^{2} - 5 p T^{3} + 1121 T^{4} - 2040 T^{5} - 44678 T^{6} + 79040 T^{7} + 1805724 T^{8} + 79040 p T^{9} - 44678 p^{2} T^{10} - 2040 p^{3} T^{11} + 1121 p^{4} T^{12} - 5 p^{6} T^{13} - 48 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 733824 p^{2} T^{10} + 12668 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 4 T - 74 T^{2} - 288 T^{3} + 4517 T^{4} + 556 p T^{5} - 11538 T^{6} - 738968 T^{7} - 2462804 T^{8} - 738968 p T^{9} - 11538 p^{2} T^{10} + 556 p^{4} T^{11} + 4517 p^{4} T^{12} - 288 p^{5} T^{13} - 74 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 194436327001344 p^{2} T^{18} + 3558033934834 p^{4} T^{20} + 54954910764 p^{6} T^{22} + 705172105 p^{8} T^{24} + 7305024 p^{10} T^{26} + 58258 p^{12} T^{28} + 324 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 3 T - 98 T^{2} - 219 T^{3} + 3523 T^{4} - 1116 T^{5} - 253856 T^{6} + 226218 T^{7} + 18909448 T^{8} + 226218 p T^{9} - 253856 p^{2} T^{10} - 1116 p^{3} T^{11} + 3523 p^{4} T^{12} - 219 p^{5} T^{13} - 98 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 3123720 p^{2} T^{10} + 35516 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 506327334867240 p^{2} T^{18} + 7291973853618 p^{4} T^{20} + 92991430988 p^{6} T^{22} + 1028964713 p^{8} T^{24} + 9549968 p^{10} T^{26} + 70130 p^{12} T^{28} + 364 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 264102902456582 p^{2} T^{18} + 3801157073266 p^{4} T^{20} + 49803739968 p^{6} T^{22} + 584536925 p^{8} T^{24} + 6035887 p^{10} T^{26} + 52152 p^{12} T^{28} + 323 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 193036342550180 p^{2} T^{18} + 2631719645962 p^{4} T^{20} + 33090030780 p^{6} T^{22} + 362561921 p^{8} T^{24} + 3796936 p^{10} T^{26} + 39450 p^{12} T^{28} + 296 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 18 T + 332 T^{2} + 3582 T^{3} + 36198 T^{4} + 3582 p T^{5} + 332 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( ( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 2325 p T^{5} + 278 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 79 | \( ( 1 + 15 T - 80 T^{2} - 1305 T^{3} + 13273 T^{4} + 59400 T^{5} - 1907150 T^{6} - 913560 T^{7} + 190569148 T^{8} - 913560 p T^{9} - 1907150 p^{2} T^{10} + 59400 p^{3} T^{11} + 13273 p^{4} T^{12} - 1305 p^{5} T^{13} - 80 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 10182388433060610 p^{2} T^{18} + 99713806040442 p^{4} T^{20} + 843016361960 p^{6} T^{22} + 6055752221 p^{8} T^{24} + 35967119 p^{10} T^{26} + 168260 p^{12} T^{28} + 559 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 16 T + 408 T^{2} + 4116 T^{3} + 56278 T^{4} + 4116 p T^{5} + 408 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 97 | \( ( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 15575820 p T^{9} - 2055806 p^{2} T^{10} - 273780 p^{3} T^{11} + 35209 p^{4} T^{12} + 1560 p^{5} T^{13} - 254 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−3.57933961320594198893280022330, −3.34563926440570693115614320553, −3.26835906253130334969557043092, −3.22691704942232860884107827576, −3.15126917820634993782343522014, −3.12938621361286864294156975158, −3.09235404359095690525709130983, −2.90364461052655166680526459242, −2.84905650215004957621079469695, −2.74904789831606887210176424309, −2.54568166440474899826445236582, −2.52935640703144750636400486476, −2.46368104814122253318788757442, −2.27448570432677294920759657740, −2.10449645220910708925499778287, −1.83115633734538758855093921274, −1.62023980263603489298278004554, −1.53582123941884095153597259643, −1.51608097079731405805175068206, −1.47155338009370854775886172198, −1.35004714164705392953881123314, −1.27143972351742740428900466521, −1.19421395709542302221689341577, −0.822201316537317741535202261830, −0.22735496729930546521910322768, 0.22735496729930546521910322768, 0.822201316537317741535202261830, 1.19421395709542302221689341577, 1.27143972351742740428900466521, 1.35004714164705392953881123314, 1.47155338009370854775886172198, 1.51608097079731405805175068206, 1.53582123941884095153597259643, 1.62023980263603489298278004554, 1.83115633734538758855093921274, 2.10449645220910708925499778287, 2.27448570432677294920759657740, 2.46368104814122253318788757442, 2.52935640703144750636400486476, 2.54568166440474899826445236582, 2.74904789831606887210176424309, 2.84905650215004957621079469695, 2.90364461052655166680526459242, 3.09235404359095690525709130983, 3.12938621361286864294156975158, 3.15126917820634993782343522014, 3.22691704942232860884107827576, 3.26835906253130334969557043092, 3.34563926440570693115614320553, 3.57933961320594198893280022330
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.