Dirichlet series
| L(s) = 1 | + 2.04e3·4-s + 1.83e4·7-s + 1.12e5·9-s + 4.30e5·11-s + 2.62e6·16-s + 6.26e6·23-s + 2.47e7·25-s + 3.76e7·28-s − 4.64e7·29-s + 2.31e8·36-s + 3.60e8·37-s + 3.21e7·43-s + 8.82e8·44-s + 5.95e8·49-s − 1.13e9·53-s + 2.07e9·63-s + 2.68e9·64-s − 2.25e9·67-s + 2.12e9·71-s + 7.91e9·77-s − 5.25e9·79-s + 8.91e9·81-s + 1.28e10·92-s + 4.86e10·99-s + 5.05e10·100-s + 7.50e10·107-s − 7.13e9·109-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.09·7-s + 1.91·9-s + 2.67·11-s + 5/2·16-s + 0.973·23-s + 2.52·25-s + 2.18·28-s − 2.26·29-s + 3.82·36-s + 5.20·37-s + 0.218·43-s + 5.34·44-s + 2.10·49-s − 2.70·53-s + 2.09·63-s + 5/2·64-s − 1.67·67-s + 1.17·71-s + 2.92·77-s − 1.70·79-s + 2.55·81-s + 1.94·92-s + 5.11·99-s + 5.05·100-s + 5.35·107-s − 0.463·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.91893\times 10^{7}\) |
| Root analytic conductor: | \(2.98244\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | induced by $\chi_{14} (1, \cdot )$ |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [5]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(59.52206142\) |
| \(L(\frac12)\) | \(\approx\) | \(59.52206142\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{9} T^{2} )^{4} \) |
| 7 | \( 1 - 18376 T - 36872252 p T^{2} - 838568440 p^{4} T^{3} + 39820013830 p^{8} T^{4} - 838568440 p^{14} T^{5} - 36872252 p^{21} T^{6} - 18376 p^{30} T^{7} + p^{40} T^{8} \) | |
| good | 3 | \( 1 - 4184 p^{3} T^{2} + 426947068 p^{2} T^{4} - 1323702184 p^{5} T^{6} - 164294104749242 p^{8} T^{8} - 1323702184 p^{25} T^{10} + 426947068 p^{42} T^{12} - 4184 p^{63} T^{14} + p^{80} T^{16} \) |
| 5 | \( 1 - 4940584 p T^{2} + 18858441490876 p^{2} T^{4} - 49698314325091017304 p^{3} T^{6} + \)\(10\!\cdots\!86\)\( p^{4} T^{8} - 49698314325091017304 p^{23} T^{10} + 18858441490876 p^{42} T^{12} - 4940584 p^{61} T^{14} + p^{80} T^{16} \) | |
| 11 | \( ( 1 - 215400 T + 44365220668 T^{2} - 1689454080938520 T^{3} + \)\(46\!\cdots\!58\)\( T^{4} - 1689454080938520 p^{10} T^{5} + 44365220668 p^{20} T^{6} - 215400 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 13 | \( 1 - 18362025320 p T^{2} + \)\(35\!\cdots\!80\)\( T^{4} - \)\(20\!\cdots\!60\)\( p T^{6} + \)\(64\!\cdots\!62\)\( T^{8} - \)\(20\!\cdots\!60\)\( p^{21} T^{10} + \)\(35\!\cdots\!80\)\( p^{40} T^{12} - 18362025320 p^{61} T^{14} + p^{80} T^{16} \) | |
| 17 | \( 1 - 1453258511624 T^{2} + \)\(49\!\cdots\!36\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(22\!\cdots\!26\)\( T^{8} - \)\(17\!\cdots\!08\)\( p^{20} T^{10} + \)\(49\!\cdots\!36\)\( p^{40} T^{12} - 1453258511624 p^{60} T^{14} + p^{80} T^{16} \) | |
| 19 | \( 1 - 20960775311432 T^{2} + \)\(21\!\cdots\!52\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{6} + \)\(87\!\cdots\!98\)\( T^{8} - \)\(14\!\cdots\!08\)\( p^{20} T^{10} + \)\(21\!\cdots\!52\)\( p^{40} T^{12} - 20960775311432 p^{60} T^{14} + p^{80} T^{16} \) | |
| 23 | \( ( 1 - 3132744 T + 114893271672028 T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(59\!\cdots\!58\)\( T^{4} - \)\(17\!\cdots\!64\)\( p^{10} T^{5} + 114893271672028 p^{20} T^{6} - 3132744 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 29 | \( ( 1 + 23215704 T + 1727350367025916 T^{2} + \)\(27\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(27\!\cdots\!08\)\( p^{10} T^{5} + 1727350367025916 p^{20} T^{6} + 23215704 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 31 | \( 1 - 5039894655794696 T^{2} + \)\(12\!\cdots\!36\)\( T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!26\)\( T^{8} - \)\(17\!\cdots\!92\)\( p^{20} T^{10} + \)\(12\!\cdots\!36\)\( p^{40} T^{12} - 5039894655794696 p^{60} T^{14} + p^{80} T^{16} \) | |
| 37 | \( ( 1 - 180466408 T + 28001115462100924 T^{2} - \)\(25\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!06\)\( T^{4} - \)\(25\!\cdots\!64\)\( p^{10} T^{5} + 28001115462100924 p^{20} T^{6} - 180466408 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 41 | \( 1 - 81468760613148296 T^{2} + \)\(31\!\cdots\!36\)\( T^{4} - \)\(73\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!46\)\( T^{8} - \)\(73\!\cdots\!12\)\( p^{20} T^{10} + \)\(31\!\cdots\!36\)\( p^{40} T^{12} - 81468760613148296 p^{60} T^{14} + p^{80} T^{16} \) | |
| 43 | \( ( 1 - 16056424 T + 70706689957015612 T^{2} - \)\(13\!\cdots\!92\)\( T^{3} + \)\(21\!\cdots\!70\)\( T^{4} - \)\(13\!\cdots\!92\)\( p^{10} T^{5} + 70706689957015612 p^{20} T^{6} - 16056424 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 47 | \( 1 - 261498778361053448 T^{2} + \)\(31\!\cdots\!32\)\( T^{4} - \)\(24\!\cdots\!72\)\( T^{6} + \)\(14\!\cdots\!18\)\( T^{8} - \)\(24\!\cdots\!72\)\( p^{20} T^{10} + \)\(31\!\cdots\!32\)\( p^{40} T^{12} - 261498778361053448 p^{60} T^{14} + p^{80} T^{16} \) | |
| 53 | \( ( 1 + 566129304 T + 432701914132831036 T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!90\)\( T^{4} + \)\(12\!\cdots\!80\)\( p^{10} T^{5} + 432701914132831036 p^{20} T^{6} + 566129304 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 59 | \( 1 - 3254755403618781128 T^{2} + \)\(49\!\cdots\!48\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{6} + \)\(28\!\cdots\!70\)\( T^{8} - \)\(46\!\cdots\!56\)\( p^{20} T^{10} + \)\(49\!\cdots\!48\)\( p^{40} T^{12} - 3254755403618781128 p^{60} T^{14} + p^{80} T^{16} \) | |
| 61 | \( 1 - 1880786966665621832 T^{2} + \)\(19\!\cdots\!32\)\( T^{4} - \)\(13\!\cdots\!68\)\( T^{6} + \)\(89\!\cdots\!98\)\( T^{8} - \)\(13\!\cdots\!68\)\( p^{20} T^{10} + \)\(19\!\cdots\!32\)\( p^{40} T^{12} - 1880786966665621832 p^{60} T^{14} + p^{80} T^{16} \) | |
| 67 | \( ( 1 + 1127371096 T + 4707206769474851836 T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!90\)\( T^{4} + \)\(18\!\cdots\!20\)\( p^{10} T^{5} + 4707206769474851836 p^{20} T^{6} + 1127371096 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 71 | \( ( 1 - 1060955592 T + 7646017944722511964 T^{2} - \)\(93\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} - \)\(93\!\cdots\!20\)\( p^{10} T^{5} + 7646017944722511964 p^{20} T^{6} - 1060955592 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 73 | \( 1 - 9548995843590486152 T^{2} + \)\(27\!\cdots\!12\)\( T^{4} + \)\(38\!\cdots\!72\)\( T^{6} - \)\(44\!\cdots\!42\)\( T^{8} + \)\(38\!\cdots\!72\)\( p^{20} T^{10} + \)\(27\!\cdots\!12\)\( p^{40} T^{12} - 9548995843590486152 p^{60} T^{14} + p^{80} T^{16} \) | |
| 79 | \( ( 1 + 2628683896 T + 18313636312198280860 T^{2} + \)\(19\!\cdots\!84\)\( T^{3} + \)\(17\!\cdots\!94\)\( T^{4} + \)\(19\!\cdots\!84\)\( p^{10} T^{5} + 18313636312198280860 p^{20} T^{6} + 2628683896 p^{30} T^{7} + p^{40} T^{8} )^{2} \) | |
| 83 | \( 1 - 93992144973599427656 T^{2} + \)\(41\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{6} + \)\(21\!\cdots\!30\)\( T^{8} - \)\(11\!\cdots\!56\)\( p^{20} T^{10} + \)\(41\!\cdots\!44\)\( p^{40} T^{12} - 93992144973599427656 p^{60} T^{14} + p^{80} T^{16} \) | |
| 89 | \( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(76\!\cdots\!60\)\( T^{4} - \)\(33\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!02\)\( T^{8} - \)\(33\!\cdots\!80\)\( p^{20} T^{10} + \)\(76\!\cdots\!60\)\( p^{40} T^{12} - \)\(12\!\cdots\!00\)\( p^{60} T^{14} + p^{80} T^{16} \) | |
| 97 | \( 1 - \)\(42\!\cdots\!60\)\( T^{2} + \)\(86\!\cdots\!80\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!58\)\( p^{2} T^{8} - \)\(11\!\cdots\!20\)\( p^{20} T^{10} + \)\(86\!\cdots\!80\)\( p^{40} T^{12} - \)\(42\!\cdots\!60\)\( p^{60} T^{14} + p^{80} T^{16} \) | |
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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
−7.36837635079970311534495687317, −6.97477001408973618948086159633, −6.58422543953362877425095432633, −6.54880411474577385854377787738, −6.43668002452953444856434340039, −6.37708958732547332061082870058, −5.81814049803714168033943631773, −5.50855271373012650310600274762, −5.38386346636256152470892424029, −4.94263848794257047145445225191, −4.39408890038465783672945511106, −4.30468473038160613552491324784, −4.30066875962567677753601434448, −4.03671868456822335296064018859, −3.41079211156682390444320424666, −3.20793651054077091309507445031, −2.74209552858410862585971067633, −2.73839477885835949782509701705, −2.01319491228743940338007891214, −1.90214209121771591660725441177, −1.46233485714256613462260377276, −1.45907204740700640589768969609, −0.939049889262976462111059464448, −0.895290693171987341449049625049, −0.60556484810660913135645690916, 0.60556484810660913135645690916, 0.895290693171987341449049625049, 0.939049889262976462111059464448, 1.45907204740700640589768969609, 1.46233485714256613462260377276, 1.90214209121771591660725441177, 2.01319491228743940338007891214, 2.73839477885835949782509701705, 2.74209552858410862585971067633, 3.20793651054077091309507445031, 3.41079211156682390444320424666, 4.03671868456822335296064018859, 4.30066875962567677753601434448, 4.30468473038160613552491324784, 4.39408890038465783672945511106, 4.94263848794257047145445225191, 5.38386346636256152470892424029, 5.50855271373012650310600274762, 5.81814049803714168033943631773, 6.37708958732547332061082870058, 6.43668002452953444856434340039, 6.54880411474577385854377787738, 6.58422543953362877425095432633, 6.97477001408973618948086159633, 7.36837635079970311534495687317
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.