Dirichlet series
| L(s) = 1 | + 4·2-s − 7·3-s + 4·4-s − 30·5-s − 28·6-s + 4·7-s + 11·9-s − 120·10-s − 28·12-s − 48·13-s + 16·14-s + 210·15-s − 109·17-s + 44·18-s + 288·19-s − 120·20-s − 28·21-s + 628·23-s + 594·25-s − 192·26-s + 91·27-s + 16·28-s + 528·29-s + 840·30-s − 522·31-s − 64·32-s − 436·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.34·3-s + 1/2·4-s − 2.68·5-s − 1.90·6-s + 0.215·7-s + 0.407·9-s − 3.79·10-s − 0.673·12-s − 1.02·13-s + 0.305·14-s + 3.61·15-s − 1.55·17-s + 0.576·18-s + 3.47·19-s − 1.34·20-s − 0.290·21-s + 5.69·23-s + 4.75·25-s − 1.44·26-s + 0.648·27-s + 0.107·28-s + 3.38·29-s + 5.11·30-s − 3.02·31-s − 0.353·32-s − 2.19·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.72763\times 10^{9}\) |
| Root analytic conductor: | \(3.77868\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 11^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(7.801330524\) |
| \(L(\frac12)\) | \(\approx\) | \(7.801330524\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 7 T + 38 T^{2} + 98 T^{3} + 62 T^{4} + 4223 T^{5} + 3691 p^{2} T^{6} + 71332 p T^{7} + 1147672 T^{8} + 71332 p^{4} T^{9} + 3691 p^{8} T^{10} + 4223 p^{9} T^{11} + 62 p^{12} T^{12} + 98 p^{15} T^{13} + 38 p^{18} T^{14} + 7 p^{21} T^{15} + p^{24} T^{16} \) |
| 5 | \( 1 + 6 p T + 306 T^{2} + 184 p T^{3} - 569 T^{4} + 17062 p T^{5} + 1402636 T^{6} + 3741048 p T^{7} + 272699281 T^{8} + 3741048 p^{4} T^{9} + 1402636 p^{6} T^{10} + 17062 p^{10} T^{11} - 569 p^{12} T^{12} + 184 p^{16} T^{13} + 306 p^{18} T^{14} + 6 p^{22} T^{15} + p^{24} T^{16} \) | |
| 7 | \( 1 - 4 T - 278 T^{2} + 1486 T^{3} - 85843 T^{4} - 84088 p T^{5} + 34111706 T^{6} + 31711312 T^{7} + 2545738687 T^{8} + 31711312 p^{3} T^{9} + 34111706 p^{6} T^{10} - 84088 p^{10} T^{11} - 85843 p^{12} T^{12} + 1486 p^{15} T^{13} - 278 p^{18} T^{14} - 4 p^{21} T^{15} + p^{24} T^{16} \) | |
| 13 | \( 1 + 48 T - 5550 T^{2} - 362926 T^{3} + 4851591 T^{4} + 1163685872 T^{5} + 49432845496 T^{6} - 1273897050480 T^{7} - 189168222289331 T^{8} - 1273897050480 p^{3} T^{9} + 49432845496 p^{6} T^{10} + 1163685872 p^{9} T^{11} + 4851591 p^{12} T^{12} - 362926 p^{15} T^{13} - 5550 p^{18} T^{14} + 48 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 + 109 T - 1340 T^{2} + 17028 T^{3} + 61854916 T^{4} + 3258771851 T^{5} + 4844670167 p T^{6} + 11257525677000 T^{7} + 1063382720825544 T^{8} + 11257525677000 p^{3} T^{9} + 4844670167 p^{7} T^{10} + 3258771851 p^{9} T^{11} + 61854916 p^{12} T^{12} + 17028 p^{15} T^{13} - 1340 p^{18} T^{14} + 109 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( 1 - 288 T + 30566 T^{2} - 95904 p T^{3} + 176552535 T^{4} - 24586907808 T^{5} + 2222669375260 T^{6} - 119753076108480 T^{7} + 6107895510065021 T^{8} - 119753076108480 p^{3} T^{9} + 2222669375260 p^{6} T^{10} - 24586907808 p^{9} T^{11} + 176552535 p^{12} T^{12} - 95904 p^{16} T^{13} + 30566 p^{18} T^{14} - 288 p^{21} T^{15} + p^{24} T^{16} \) | |
| 23 | \( ( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} - 314 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 29 | \( 1 - 528 T + 82486 T^{2} + 8724144 T^{3} - 5230024245 T^{4} + 769204136712 T^{5} - 8600199101200 T^{6} - 17285931767133480 T^{7} + 3853264451262183881 T^{8} - 17285931767133480 p^{3} T^{9} - 8600199101200 p^{6} T^{10} + 769204136712 p^{9} T^{11} - 5230024245 p^{12} T^{12} + 8724144 p^{15} T^{13} + 82486 p^{18} T^{14} - 528 p^{21} T^{15} + p^{24} T^{16} \) | |
| 31 | \( 1 + 522 T + 160492 T^{2} + 35181052 T^{3} + 6294650003 T^{4} + 763981141902 T^{5} + 33910403803870 T^{6} - 9279992614495420 T^{7} - 2496218287452452949 T^{8} - 9279992614495420 p^{3} T^{9} + 33910403803870 p^{6} T^{10} + 763981141902 p^{9} T^{11} + 6294650003 p^{12} T^{12} + 35181052 p^{15} T^{13} + 160492 p^{18} T^{14} + 522 p^{21} T^{15} + p^{24} T^{16} \) | |
| 37 | \( 1 + 406 T - 5318 T^{2} - 27663524 T^{3} - 3521516073 T^{4} + 1758227311414 T^{5} + 498571849360436 T^{6} - 45032760859024008 T^{7} - 32601870976696136623 T^{8} - 45032760859024008 p^{3} T^{9} + 498571849360436 p^{6} T^{10} + 1758227311414 p^{9} T^{11} - 3521516073 p^{12} T^{12} - 27663524 p^{15} T^{13} - 5318 p^{18} T^{14} + 406 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( 1 + 329 T - 55160 T^{2} - 17291460 T^{3} + 3867540040 T^{4} + 810769263487 T^{5} + 94427755341103 T^{6} - 42504597447646080 T^{7} - 34767930814290379080 T^{8} - 42504597447646080 p^{3} T^{9} + 94427755341103 p^{6} T^{10} + 810769263487 p^{9} T^{11} + 3867540040 p^{12} T^{12} - 17291460 p^{15} T^{13} - 55160 p^{18} T^{14} + 329 p^{21} T^{15} + p^{24} T^{16} \) | |
| 43 | \( ( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} - 721 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 47 | \( 1 - 666 T + 215830 T^{2} - 1878276 p T^{3} + 49867598541 T^{4} - 15263379264354 T^{5} + 2694775262754914 T^{6} - 1173324858240942840 T^{7} + \)\(55\!\cdots\!99\)\( T^{8} - 1173324858240942840 p^{3} T^{9} + 2694775262754914 p^{6} T^{10} - 15263379264354 p^{9} T^{11} + 49867598541 p^{12} T^{12} - 1878276 p^{16} T^{13} + 215830 p^{18} T^{14} - 666 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 414 T + 111938 T^{2} - 31720518 T^{3} + 42174716831 T^{4} - 4836347208574 T^{5} - 2624686559799020 T^{6} + 993826932968752496 T^{7} + \)\(38\!\cdots\!49\)\( T^{8} + 993826932968752496 p^{3} T^{9} - 2624686559799020 p^{6} T^{10} - 4836347208574 p^{9} T^{11} + 42174716831 p^{12} T^{12} - 31720518 p^{15} T^{13} + 111938 p^{18} T^{14} - 414 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( 1 + 888 T - 72874 T^{2} - 366469944 T^{3} - 125594098825 T^{4} + 29314994105528 T^{5} + 26622351545153740 T^{6} + 1700698924437048800 T^{7} - \)\(25\!\cdots\!59\)\( T^{8} + 1700698924437048800 p^{3} T^{9} + 26622351545153740 p^{6} T^{10} + 29314994105528 p^{9} T^{11} - 125594098825 p^{12} T^{12} - 366469944 p^{15} T^{13} - 72874 p^{18} T^{14} + 888 p^{21} T^{15} + p^{24} T^{16} \) | |
| 61 | \( 1 + 302 T - 374758 T^{2} - 265079458 T^{3} + 17172817983 T^{4} + 53588613835342 T^{5} + 17468090773098500 T^{6} - 4210498173089967520 T^{7} - \)\(41\!\cdots\!19\)\( T^{8} - 4210498173089967520 p^{3} T^{9} + 17468090773098500 p^{6} T^{10} + 53588613835342 p^{9} T^{11} + 17172817983 p^{12} T^{12} - 265079458 p^{15} T^{13} - 374758 p^{18} T^{14} + 302 p^{21} T^{15} + p^{24} T^{16} \) | |
| 67 | \( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 71 | \( 1 - 1090 T - 143028 T^{2} + 705054350 T^{3} - 262816671537 T^{4} - 195394750802610 T^{5} + 164095529988407974 T^{6} + 23390610332557696900 T^{7} - \)\(62\!\cdots\!45\)\( T^{8} + 23390610332557696900 p^{3} T^{9} + 164095529988407974 p^{6} T^{10} - 195394750802610 p^{9} T^{11} - 262816671537 p^{12} T^{12} + 705054350 p^{15} T^{13} - 143028 p^{18} T^{14} - 1090 p^{21} T^{15} + p^{24} T^{16} \) | |
| 73 | \( 1 - 253 T + 242148 T^{2} + 153484248 T^{3} + 27413114052 T^{4} + 97831236100713 T^{5} + 36243442544494539 T^{6} + 8871101046105798936 T^{7} + \)\(23\!\cdots\!32\)\( T^{8} + 8871101046105798936 p^{3} T^{9} + 36243442544494539 p^{6} T^{10} + 97831236100713 p^{9} T^{11} + 27413114052 p^{12} T^{12} + 153484248 p^{15} T^{13} + 242148 p^{18} T^{14} - 253 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( 1 + 674 T - 776972 T^{2} - 654255424 T^{3} + 174326887083 T^{4} + 318069877959854 T^{5} + 120607404735943670 T^{6} - 80049450610837979100 T^{7} - \)\(13\!\cdots\!89\)\( T^{8} - 80049450610837979100 p^{3} T^{9} + 120607404735943670 p^{6} T^{10} + 318069877959854 p^{9} T^{11} + 174326887083 p^{12} T^{12} - 654255424 p^{15} T^{13} - 776972 p^{18} T^{14} + 674 p^{21} T^{15} + p^{24} T^{16} \) | |
| 83 | \( 1 + 428 T + 177930 T^{2} - 134660776 T^{3} - 247636118949 T^{4} - 4577801916276 p T^{5} + 90010500467785276 T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!89\)\( T^{8} + \)\(15\!\cdots\!00\)\( p^{3} T^{9} + 90010500467785276 p^{6} T^{10} - 4577801916276 p^{10} T^{11} - 247636118949 p^{12} T^{12} - 134660776 p^{15} T^{13} + 177930 p^{18} T^{14} + 428 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 97 | \( 1 - 3012 T + 5947742 T^{2} - 10226856716 T^{3} + 16448850705271 T^{4} - 21915763249625172 T^{5} + 26024459406245646260 T^{6} - \)\(28\!\cdots\!48\)\( T^{7} + \)\(29\!\cdots\!09\)\( T^{8} - \)\(28\!\cdots\!48\)\( p^{3} T^{9} + 26024459406245646260 p^{6} T^{10} - 21915763249625172 p^{9} T^{11} + 16448850705271 p^{12} T^{12} - 10226856716 p^{15} T^{13} + 5947742 p^{18} T^{14} - 3012 p^{21} T^{15} + p^{24} T^{16} \) | |
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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
−4.93891363600133483393325247540, −4.75242945300469852121817538979, −4.65690659872422110053143779107, −4.62791903579625435160938100921, −4.56765584689381076337790456508, −4.31945020543645110114480543182, −3.97617781387884863247953477807, −3.83589917980480301124317244913, −3.62456290783945569124048441430, −3.48862108786274895017614488872, −3.47110082477593793495540802700, −3.29191306668188294758445752166, −2.96370688976715080579972796161, −2.94309421408336669865945261210, −2.68034655172719926596001637579, −2.43708678727578445978575942477, −2.36535097834120101568169491666, −2.23288451373228201405013111681, −1.37686211962384787547635006558, −1.20576974971142512780797088785, −1.18036045484269135614063586473, −0.936909681400372146118711150338, −0.60875905526457326674449107027, −0.52555969208373491429745053604, −0.33094090934217748989975452685, 0.33094090934217748989975452685, 0.52555969208373491429745053604, 0.60875905526457326674449107027, 0.936909681400372146118711150338, 1.18036045484269135614063586473, 1.20576974971142512780797088785, 1.37686211962384787547635006558, 2.23288451373228201405013111681, 2.36535097834120101568169491666, 2.43708678727578445978575942477, 2.68034655172719926596001637579, 2.94309421408336669865945261210, 2.96370688976715080579972796161, 3.29191306668188294758445752166, 3.47110082477593793495540802700, 3.48862108786274895017614488872, 3.62456290783945569124048441430, 3.83589917980480301124317244913, 3.97617781387884863247953477807, 4.31945020543645110114480543182, 4.56765584689381076337790456508, 4.62791903579625435160938100921, 4.65690659872422110053143779107, 4.75242945300469852121817538979, 4.93891363600133483393325247540
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.