Dirichlet series
| L(s) = 1 | + 5·4-s + 6·7-s + 12·13-s + 14·16-s − 24·19-s + 20·25-s + 30·28-s − 20·31-s + 4·37-s + 20·43-s + 11·49-s + 60·52-s − 36·61-s + 37·64-s + 18·67-s − 32·73-s − 120·76-s + 32·79-s + 72·91-s + 28·97-s + 100·100-s − 82·103-s + 16·109-s + 84·112-s + 53·121-s − 100·124-s + 127-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 2.26·7-s + 3.32·13-s + 7/2·16-s − 5.50·19-s + 4·25-s + 5.66·28-s − 3.59·31-s + 0.657·37-s + 3.04·43-s + 11/7·49-s + 8.32·52-s − 4.60·61-s + 37/8·64-s + 2.19·67-s − 3.74·73-s − 13.7·76-s + 3.60·79-s + 7.54·91-s + 2.84·97-s + 10·100-s − 8.07·103-s + 1.53·109-s + 7.93·112-s + 4.81·121-s − 8.98·124-s + 0.0887·127-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{64} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.11718\times 10^{10}\) |
| Root analytic conductor: | \(2.12779\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{64} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5965931511\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5965931511\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 - 3 T + 8 T^{2} - 33 T^{3} + 123 T^{4} - 33 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 2 | \( ( 1 - p^{2} T^{2} + 7 p T^{4} - 39 T^{6} + 77 T^{8} - 39 p^{2} T^{10} + 7 p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16} )( 1 - T^{2} - 7 T^{4} + 3 T^{6} + 29 T^{8} + 3 p^{2} T^{10} - 7 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} ) \) |
| 5 | \( 1 - 4 p T^{2} + 158 T^{4} - 1006 T^{6} + 8963 T^{8} - 61399 T^{10} + 293727 T^{12} - 1729137 T^{14} + 10444771 T^{16} - 1729137 p^{2} T^{18} + 293727 p^{4} T^{20} - 61399 p^{6} T^{22} + 8963 p^{8} T^{24} - 1006 p^{10} T^{26} + 158 p^{12} T^{28} - 4 p^{15} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 53 T^{2} + 1301 T^{4} - 2210 p T^{6} + 438422 T^{8} - 6939277 T^{10} + 93358746 T^{12} - 1178831562 T^{14} + 13833581305 T^{16} - 1178831562 p^{2} T^{18} + 93358746 p^{4} T^{20} - 6939277 p^{6} T^{22} + 438422 p^{8} T^{24} - 2210 p^{11} T^{26} + 1301 p^{12} T^{28} - 53 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 - 3 T + 20 T^{2} - 15 T^{3} + 231 T^{4} - 15 p T^{5} + 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 17 | \( 1 - 82 T^{2} + 3429 T^{4} - 97574 T^{6} + 2107148 T^{8} - 2059884 p T^{10} + 442861075 T^{12} - 4531943836 T^{14} + 55716215391 T^{16} - 4531943836 p^{2} T^{18} + 442861075 p^{4} T^{20} - 2059884 p^{7} T^{22} + 2107148 p^{8} T^{24} - 97574 p^{10} T^{26} + 3429 p^{12} T^{28} - 82 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 12 T + 58 T^{2} + 234 T^{3} + 853 T^{4} - 225 T^{5} - 20609 T^{6} - 7833 p T^{7} - 769679 T^{8} - 7833 p^{2} T^{9} - 20609 p^{2} T^{10} - 225 p^{3} T^{11} + 853 p^{4} T^{12} + 234 p^{5} T^{13} + 58 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 52 T^{2} + 150 T^{4} + 14626 T^{6} + 27145 p T^{8} - 12478623 T^{10} - 438951857 T^{12} + 2082575939 T^{14} + 287649733203 T^{16} + 2082575939 p^{2} T^{18} - 438951857 p^{4} T^{20} - 12478623 p^{6} T^{22} + 27145 p^{9} T^{24} + 14626 p^{10} T^{26} + 150 p^{12} T^{28} - 52 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 140 T^{2} + 10010 T^{4} + 467184 T^{6} + 15748739 T^{8} + 467184 p^{2} T^{10} + 10010 p^{4} T^{12} + 140 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 10 T - 19 T^{2} - 106 T^{3} + 2876 T^{4} + 1100 T^{5} - 124053 T^{6} - 7284 p T^{7} + 48289 p T^{8} - 7284 p^{2} T^{9} - 124053 p^{2} T^{10} + 1100 p^{3} T^{11} + 2876 p^{4} T^{12} - 106 p^{5} T^{13} - 19 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 2 T - 100 T^{2} - 70 T^{3} + 5747 T^{4} + 9995 T^{5} - 228417 T^{6} - 195273 T^{7} + 7862563 T^{8} - 195273 p T^{9} - 228417 p^{2} T^{10} + 9995 p^{3} T^{11} + 5747 p^{4} T^{12} - 70 p^{5} T^{13} - 100 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 137 T^{2} + 9536 T^{4} + 501891 T^{6} + 22487063 T^{8} + 501891 p^{2} T^{10} + 9536 p^{4} T^{12} + 137 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 5 T + 94 T^{2} - 197 T^{3} + 4069 T^{4} - 197 p T^{5} + 94 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 - 187 T^{2} + 17658 T^{4} - 1097501 T^{6} + 46137122 T^{8} - 883107198 T^{10} - 54170427851 T^{12} + 6798567948140 T^{14} - 400845896701047 T^{16} + 6798567948140 p^{2} T^{18} - 54170427851 p^{4} T^{20} - 883107198 p^{6} T^{22} + 46137122 p^{8} T^{24} - 1097501 p^{10} T^{26} + 17658 p^{12} T^{28} - 187 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 136 T^{2} + 11898 T^{4} - 982454 T^{6} + 65260463 T^{8} - 3934825011 T^{10} + 227355327775 T^{12} - 12350790854677 T^{14} + 663083591655843 T^{16} - 12350790854677 p^{2} T^{18} + 227355327775 p^{4} T^{20} - 3934825011 p^{6} T^{22} + 65260463 p^{8} T^{24} - 982454 p^{10} T^{26} + 11898 p^{12} T^{28} - 136 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 412 T^{2} + 92526 T^{4} - 14617286 T^{6} + 1798129247 T^{8} - 181339187307 T^{10} + 15440137281895 T^{12} - 1128686492503633 T^{14} + 71461191506541507 T^{16} - 1128686492503633 p^{2} T^{18} + 15440137281895 p^{4} T^{20} - 181339187307 p^{6} T^{22} + 1798129247 p^{8} T^{24} - 14617286 p^{10} T^{26} + 92526 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 18 T + 67 T^{2} - 138 T^{3} + 2530 T^{4} + 14328 T^{5} - 334169 T^{6} - 4214598 T^{7} - 32506397 T^{8} - 4214598 p T^{9} - 334169 p^{2} T^{10} + 14328 p^{3} T^{11} + 2530 p^{4} T^{12} - 138 p^{5} T^{13} + 67 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 9 T - 53 T^{2} - 228 T^{3} + 7276 T^{4} + 14733 T^{5} + 305920 T^{6} - 2410548 T^{7} - 28627265 T^{8} - 2410548 p T^{9} + 305920 p^{2} T^{10} + 14733 p^{3} T^{11} + 7276 p^{4} T^{12} - 228 p^{5} T^{13} - 53 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 211 T^{2} + 22123 T^{4} + 2180284 T^{6} + 186777883 T^{8} + 2180284 p^{2} T^{10} + 22123 p^{4} T^{12} + 211 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 16 T - 88 T^{2} - 1570 T^{3} + 22199 T^{4} + 209489 T^{5} - 1557975 T^{6} - 2216535 T^{7} + 190716007 T^{8} - 2216535 p T^{9} - 1557975 p^{2} T^{10} + 209489 p^{3} T^{11} + 22199 p^{4} T^{12} - 1570 p^{5} T^{13} - 88 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 16 T - 126 T^{2} + 1486 T^{3} + 37337 T^{4} - 221643 T^{5} - 3862157 T^{6} + 1655231 T^{7} + 447741333 T^{8} + 1655231 p T^{9} - 3862157 p^{2} T^{10} - 221643 p^{3} T^{11} + 37337 p^{4} T^{12} + 1486 p^{5} T^{13} - 126 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 553 T^{2} + 141124 T^{4} + 21681727 T^{6} + 2192243959 T^{8} + 21681727 p^{2} T^{10} + 141124 p^{4} T^{12} + 553 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 4 T^{2} + 10610 T^{4} - 307510 T^{6} - 23884909 T^{8} - 1439051407 T^{10} + 120404885691 T^{12} + 39552324252603 T^{14} + 7282335141246355 T^{16} + 39552324252603 p^{2} T^{18} + 120404885691 p^{4} T^{20} - 1439051407 p^{6} T^{22} - 23884909 p^{8} T^{24} - 307510 p^{10} T^{26} + 10610 p^{12} T^{28} + 4 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 7 T + 248 T^{2} - 1461 T^{3} + 30281 T^{4} - 1461 p T^{5} + 248 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
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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
−2.87320330182179210669973616218, −2.70012229300031914122176498920, −2.67201319597907113378013633941, −2.62572550676365949860183203483, −2.48345813995628448060801416549, −2.46050044921994584994125605386, −2.33471820724337585353745447734, −2.28127552690483562784297006027, −2.25842871290572638777445837503, −2.10520050949947398169414904179, −2.00578508381240041867726752870, −1.83325011591643007466950379242, −1.79159921881837028356358531596, −1.67962559926462854111824903477, −1.64686653680827616898102988237, −1.62563614773303150125020108312, −1.47614002420573172837877414546, −1.39809003884794846660408727703, −1.27029612115338170958390381798, −1.22478246562665312628652339678, −0.848103981298735577522859941001, −0.70758864984472242576688756720, −0.69491116027861631629233463245, −0.68619437825336471599434631607, −0.02692397158543296147048926901, 0.02692397158543296147048926901, 0.68619437825336471599434631607, 0.69491116027861631629233463245, 0.70758864984472242576688756720, 0.848103981298735577522859941001, 1.22478246562665312628652339678, 1.27029612115338170958390381798, 1.39809003884794846660408727703, 1.47614002420573172837877414546, 1.62563614773303150125020108312, 1.64686653680827616898102988237, 1.67962559926462854111824903477, 1.79159921881837028356358531596, 1.83325011591643007466950379242, 2.00578508381240041867726752870, 2.10520050949947398169414904179, 2.25842871290572638777445837503, 2.28127552690483562784297006027, 2.33471820724337585353745447734, 2.46050044921994584994125605386, 2.48345813995628448060801416549, 2.62572550676365949860183203483, 2.67201319597907113378013633941, 2.70012229300031914122176498920, 2.87320330182179210669973616218
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.