Dirichlet series
| L(s) = 1 | + 5·4-s − 6·13-s + 14·16-s − 24·19-s − 40·25-s − 20·31-s + 4·37-s − 10·43-s + 2·49-s − 30·52-s − 36·61-s + 37·64-s + 18·67-s − 32·73-s − 120·76-s + 32·79-s − 14·97-s − 200·100-s + 164·103-s + 16·109-s − 106·121-s − 100·124-s + 127-s + 131-s + 137-s + 139-s + 20·148-s + ⋯ |
| L(s) = 1 | + 5/2·4-s − 1.66·13-s + 7/2·16-s − 5.50·19-s − 8·25-s − 3.59·31-s + 0.657·37-s − 1.52·43-s + 2/7·49-s − 4.16·52-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 − 1.42·97-s − 20·100-s + 16.1·103-s + 1.53·109-s − 9.63·121-s − 8.98·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.64·148-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 | \( ( 1 - T^{2} - 18 T^{3} + 15 T^{4} - 18 p T^{5} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 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 | \( ( 1 + 4 p T^{2} + 242 T^{4} + 1917 T^{6} + 11261 T^{8} + 1917 p^{2} T^{10} + 242 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 11 | \( ( 1 + 53 T^{2} + 1508 T^{4} + 27807 T^{6} + 361871 T^{8} + 27807 p^{2} T^{10} + 1508 p^{4} T^{12} + 53 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 9 T + 20 T^{2} + 99 T^{3} - 705 T^{4} + 99 p T^{5} + 20 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 12 T + 77 T^{2} + 384 T^{3} + 1557 T^{4} + 384 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 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 | \( ( 1 + 52 T^{2} + 2554 T^{4} + 73717 T^{6} + 2065297 T^{8} + 73717 p^{2} T^{10} + 2554 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 140 T^{2} + 9590 T^{4} - 467032 T^{6} + 19045601 T^{8} - 659766664 T^{10} + 19769533926 T^{12} - 566124824436 T^{14} + 16412572563364 T^{16} - 566124824436 p^{2} T^{18} + 19769533926 p^{4} T^{20} - 659766664 p^{6} T^{22} + 19045601 p^{8} T^{24} - 467032 p^{10} T^{26} + 9590 p^{12} T^{28} - 140 p^{14} T^{30} + p^{16} T^{32} \) | |
| 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 | \( 1 - 137 T^{2} + 9233 T^{4} - 302650 T^{6} - 310834 T^{8} + 531743915 T^{10} - 24233134722 T^{12} + 462158108610 T^{14} - 4403704880711 T^{16} + 462158108610 p^{2} T^{18} - 24233134722 p^{4} T^{20} + 531743915 p^{6} T^{22} - 310834 p^{8} T^{24} - 302650 p^{10} T^{26} + 9233 p^{12} T^{28} - 137 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 5 T - 69 T^{2} + 76 T^{3} + 3782 T^{4} - 13701 T^{5} - 38420 T^{6} + 475610 T^{7} - 893241 T^{8} + 475610 p T^{9} - 38420 p^{2} T^{10} - 13701 p^{3} T^{11} + 3782 p^{4} T^{12} + 76 p^{5} T^{13} - 69 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 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 | \( 1 - 553 T^{2} + 164685 T^{4} - 34678118 T^{6} + 5733744386 T^{8} - 784555639797 T^{10} + 91514989649662 T^{12} - 9258064735293718 T^{14} + 819964567525461705 T^{16} - 9258064735293718 p^{2} T^{18} + 91514989649662 p^{4} T^{20} - 784555639797 p^{6} T^{22} + 5733744386 p^{8} T^{24} - 34678118 p^{10} T^{26} + 164685 p^{12} T^{28} - 553 p^{14} T^{30} + p^{16} T^{32} \) | |
| 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 | \( ( 1 + 7 T - 199 T^{2} - 1186 T^{3} + 20996 T^{4} + 80111 T^{5} - 1700322 T^{6} - 3328356 T^{7} + 132111961 T^{8} - 3328356 p T^{9} - 1700322 p^{2} T^{10} + 80111 p^{3} T^{11} + 20996 p^{4} T^{12} - 1186 p^{5} T^{13} - 199 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.79824520569245795073051890212, −2.67171856088688898026743361910, −2.66073305498472674897598189038, −2.60377303875470019476301133769, −2.59445264740258704972782238107, −2.50743919656148616054693632932, −2.22175572953440850901330972208, −2.16232306959341417980756373950, −2.10628462665347109519396828305, −2.08912135082647872221713999310, −1.98978296179666246067422445463, −1.91739540594862740447902208929, −1.85326696256219393281722831693, −1.83920307234750540068447894502, −1.81987741576025857086640531223, −1.77583265404332706531912237190, −1.59721895427643320941095959631, −1.50343697198153414484238088087, −1.21787758154654761389105972818, −1.20521604460384562054738625045, −0.74947522503519489732502514290, −0.63911907132581200516920499336, −0.43598902480773177547188798239, −0.17216227046461902905540869565, −0.16328639232139906382655753384, 0.16328639232139906382655753384, 0.17216227046461902905540869565, 0.43598902480773177547188798239, 0.63911907132581200516920499336, 0.74947522503519489732502514290, 1.20521604460384562054738625045, 1.21787758154654761389105972818, 1.50343697198153414484238088087, 1.59721895427643320941095959631, 1.77583265404332706531912237190, 1.81987741576025857086640531223, 1.83920307234750540068447894502, 1.85326696256219393281722831693, 1.91739540594862740447902208929, 1.98978296179666246067422445463, 2.08912135082647872221713999310, 2.10628462665347109519396828305, 2.16232306959341417980756373950, 2.22175572953440850901330972208, 2.50743919656148616054693632932, 2.59445264740258704972782238107, 2.60377303875470019476301133769, 2.66073305498472674897598189038, 2.67171856088688898026743361910, 2.79824520569245795073051890212
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.