Dirichlet series
| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s − 7·9-s − 11-s + 2·12-s − 10·17-s − 14·18-s − 15·19-s − 2·22-s − 8·25-s − 20·27-s + 23·29-s + 13·31-s − 2·32-s − 2·33-s − 20·34-s − 7·36-s + 6·37-s − 30·38-s + 2·41-s − 44-s + 10·47-s − 16·49-s − 16·50-s − 20·51-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s − 7/3·9-s − 0.301·11-s + 0.577·12-s − 2.42·17-s − 3.29·18-s − 3.44·19-s − 0.426·22-s − 8/5·25-s − 3.84·27-s + 4.27·29-s + 2.33·31-s − 0.353·32-s − 0.348·33-s − 3.42·34-s − 7/6·36-s + 0.986·37-s − 4.86·38-s + 0.312·41-s − 0.150·44-s + 1.45·47-s − 2.28·49-s − 2.26·50-s − 2.80·51-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00595070\) |
| Root analytic conductor: | \(0.725956\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.8313755423\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8313755423\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 3 | \( ( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | |
| 11 | \( 1 + T + 15 T^{2} - p T^{3} + 104 T^{4} - p^{2} T^{5} + 15 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 + 8 T^{2} + 2 p T^{3} + 29 T^{4} + 16 p T^{5} + 112 T^{6} + 28 p^{2} T^{7} + 221 T^{8} + 28 p^{3} T^{9} + 112 p^{2} T^{10} + 16 p^{4} T^{11} + 29 p^{4} T^{12} + 2 p^{6} T^{13} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 16 T^{2} + 10 T^{3} + 87 T^{4} + 160 T^{5} - 82 T^{6} + 1720 T^{7} - 3505 T^{8} + 1720 p T^{9} - 82 p^{2} T^{10} + 160 p^{3} T^{11} + 87 p^{4} T^{12} + 10 p^{5} T^{13} + 16 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 + 18 T^{2} + 10 T^{3} + 135 T^{4} + 180 T^{5} + 1568 T^{6} + 980 p T^{7} + 13729 T^{8} + 980 p^{2} T^{9} + 1568 p^{2} T^{10} + 180 p^{3} T^{11} + 135 p^{4} T^{12} + 10 p^{5} T^{13} + 18 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 10 T + 26 T^{2} + 40 T^{3} + 467 T^{4} + 3290 T^{5} + 19488 T^{6} + 56400 T^{7} + 71725 T^{8} + 56400 p T^{9} + 19488 p^{2} T^{10} + 3290 p^{3} T^{11} + 467 p^{4} T^{12} + 40 p^{5} T^{13} + 26 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 15 T + 138 T^{2} + 970 T^{3} + 5778 T^{4} + 1605 p T^{5} + 154091 T^{6} + 736340 T^{7} + 3317800 T^{8} + 736340 p T^{9} + 154091 p^{2} T^{10} + 1605 p^{4} T^{11} + 5778 p^{4} T^{12} + 970 p^{5} T^{13} + 138 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 96 T^{2} + 5212 T^{4} - 191648 T^{6} + 5108870 T^{8} - 191648 p^{2} T^{10} + 5212 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \) | |
| 29 | \( 1 - 23 T + 270 T^{2} - 2180 T^{3} + 12870 T^{4} - 47489 T^{5} + 12147 T^{6} + 1202500 T^{7} - 9195720 T^{8} + 1202500 p T^{9} + 12147 p^{2} T^{10} - 47489 p^{3} T^{11} + 12870 p^{4} T^{12} - 2180 p^{5} T^{13} + 270 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 13 T + 106 T^{2} - 872 T^{3} + 6586 T^{4} - 40319 T^{5} + 251629 T^{6} - 1513756 T^{7} + 8591372 T^{8} - 1513756 p T^{9} + 251629 p^{2} T^{10} - 40319 p^{3} T^{11} + 6586 p^{4} T^{12} - 872 p^{5} T^{13} + 106 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T - 42 T^{2} + 498 T^{3} - 789 T^{4} - 26526 T^{5} + 159760 T^{6} + 440664 T^{7} - 7027551 T^{8} + 440664 p T^{9} + 159760 p^{2} T^{10} - 26526 p^{3} T^{11} - 789 p^{4} T^{12} + 498 p^{5} T^{13} - 42 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 2 T - 114 T^{2} + 132 T^{3} + 3471 T^{4} - 3306 T^{5} + 199344 T^{6} - 1544 T^{7} - 17114163 T^{8} - 1544 p T^{9} + 199344 p^{2} T^{10} - 3306 p^{3} T^{11} + 3471 p^{4} T^{12} + 132 p^{5} T^{13} - 114 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 231 T^{2} + 27167 T^{4} - 2024653 T^{6} + 103981560 T^{8} - 2024653 p^{2} T^{10} + 27167 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 86 T^{2} + 380 T^{3} - 6733 T^{4} + 69970 T^{5} - 65532 T^{6} - 2017800 T^{7} + 29345725 T^{8} - 2017800 p T^{9} - 65532 p^{2} T^{10} + 69970 p^{3} T^{11} - 6733 p^{4} T^{12} + 380 p^{5} T^{13} + 86 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 15 T + 206 T^{2} - 2820 T^{3} + 27882 T^{4} - 271485 T^{5} + 2462023 T^{6} - 19707960 T^{7} + 150191000 T^{8} - 19707960 p T^{9} + 2462023 p^{2} T^{10} - 271485 p^{3} T^{11} + 27882 p^{4} T^{12} - 2820 p^{5} T^{13} + 206 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 25 T + 300 T^{2} + 1010 T^{3} - 15456 T^{4} - 230275 T^{5} - 679305 T^{6} + 11567740 T^{7} + 158967216 T^{8} + 11567740 p T^{9} - 679305 p^{2} T^{10} - 230275 p^{3} T^{11} - 15456 p^{4} T^{12} + 1010 p^{5} T^{13} + 300 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 10 T + 234 T^{2} + 1560 T^{3} + 20515 T^{4} + 80630 T^{5} + 839156 T^{6} + 234480 T^{7} + 25957949 T^{8} + 234480 p T^{9} + 839156 p^{2} T^{10} + 80630 p^{3} T^{11} + 20515 p^{4} T^{12} + 1560 p^{5} T^{13} + 234 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + T + 119 T^{2} - 83 T^{3} + 10044 T^{4} - 83 p T^{5} + 119 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( ( 1 - 20 T + 191 T^{2} - 830 T^{3} + 3701 T^{4} - 830 p T^{5} + 191 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( 1 - 5 T + 96 T^{2} - 220 T^{3} + 10752 T^{4} - 475 p T^{5} + 1158063 T^{6} - 2376560 T^{7} + 65290040 T^{8} - 2376560 p T^{9} + 1158063 p^{2} T^{10} - 475 p^{4} T^{11} + 10752 p^{4} T^{12} - 220 p^{5} T^{13} + 96 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 10 T + 110 T^{2} + 580 T^{3} - 1331 T^{4} + 50110 T^{5} + 938610 T^{6} - 2546560 T^{7} + 94198811 T^{8} - 2546560 p T^{9} + 938610 p^{2} T^{10} + 50110 p^{3} T^{11} - 1331 p^{4} T^{12} + 580 p^{5} T^{13} + 110 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 21 T + 86 T^{2} - 846 T^{3} - 3342 T^{4} + 29181 T^{5} - 520301 T^{6} - 8399448 T^{7} - 56629312 T^{8} - 8399448 p T^{9} - 520301 p^{2} T^{10} + 29181 p^{3} T^{11} - 3342 p^{4} T^{12} - 846 p^{5} T^{13} + 86 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 199 T^{2} + 19855 T^{4} - 1662361 T^{6} + 149152624 T^{8} - 1662361 p^{2} T^{10} + 19855 p^{4} T^{12} - 199 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 9 T + 88 T^{2} - 1068 T^{3} + 25336 T^{4} - 153839 T^{5} + 668695 T^{6} - 11831104 T^{7} + 232073144 T^{8} - 11831104 p T^{9} + 668695 p^{2} T^{10} - 153839 p^{3} T^{11} + 25336 p^{4} T^{12} - 1068 p^{5} T^{13} + 88 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.21721794950996030856036161896, −6.80294551433325624591476056817, −6.50768885062979735911828531145, −6.45412151110563902973231865948, −6.35211795798774178720427325147, −6.24999259965489034782801917360, −6.21735922517882028244322826516, −6.18434389903863806633736044042, −5.74253748640372184863498487386, −5.35047617730000454995995473419, −5.10945606701880016843023575444, −5.05367016906947677232204408446, −4.78522636852462276319893794426, −4.74501425401033814928277335406, −4.35525222567087590118200143913, −4.28287485542534995594607554011, −3.96031595649973065902395205791, −3.91521803089721800676724414689, −3.62268449802275999316575604826, −3.09194171665896979911122556877, −2.81831191006726850516452605326, −2.74378335413805103047111511603, −2.42739768799940196490023818891, −2.35287665454615504372213505347, −1.98224273110715892290112230774, 1.98224273110715892290112230774, 2.35287665454615504372213505347, 2.42739768799940196490023818891, 2.74378335413805103047111511603, 2.81831191006726850516452605326, 3.09194171665896979911122556877, 3.62268449802275999316575604826, 3.91521803089721800676724414689, 3.96031595649973065902395205791, 4.28287485542534995594607554011, 4.35525222567087590118200143913, 4.74501425401033814928277335406, 4.78522636852462276319893794426, 5.05367016906947677232204408446, 5.10945606701880016843023575444, 5.35047617730000454995995473419, 5.74253748640372184863498487386, 6.18434389903863806633736044042, 6.21735922517882028244322826516, 6.24999259965489034782801917360, 6.35211795798774178720427325147, 6.45412151110563902973231865948, 6.50768885062979735911828531145, 6.80294551433325624591476056817, 7.21721794950996030856036161896
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.