Dirichlet series
| L(s) = 1 | − 1.03e6·4-s + 3.22e8·7-s − 1.39e11·13-s + 6.88e11·16-s + 2.34e12·19-s − 9.62e13·25-s − 3.34e14·28-s − 4.60e14·31-s − 1.50e15·37-s + 1.24e15·43-s + 5.86e16·49-s + 1.44e17·52-s + 1.12e17·61-s − 2.88e17·64-s − 1.51e17·67-s − 7.36e17·73-s − 2.42e18·76-s − 1.86e18·79-s − 4.51e19·91-s − 9.32e18·97-s + 9.96e19·100-s − 1.51e19·103-s − 3.95e19·109-s + 2.22e20·112-s − 2.49e20·121-s + 4.77e20·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1.97·4-s + 3.02·7-s − 3.65·13-s + 2.50·16-s + 1.66·19-s − 5.04·25-s − 5.97·28-s − 3.13·31-s − 1.89·37-s + 0.376·43-s + 36/7·49-s + 7.22·52-s + 1.22·61-s − 2.00·64-s − 0.680·67-s − 1.46·73-s − 3.29·76-s − 1.75·79-s − 11.0·91-s − 1.24·97-s + 9.96·100-s − 1.14·103-s − 1.74·109-s + 7.57·112-s − 4.07·121-s + 6.18·124-s + 5.04·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+19/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.86477\times 10^{17}\) |
| Root analytic conductor: | \(12.0064\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [19/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | 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 | \( ( 1 - p^{9} T )^{8} \) | |
| good | 2 | \( 1 + 258943 p^{2} T^{2} + 1499685103 p^{8} T^{4} - 410605443665 p^{16} T^{6} - 1168378983796931 p^{26} T^{8} - 410605443665 p^{54} T^{10} + 1499685103 p^{84} T^{12} + 258943 p^{116} T^{14} + p^{152} T^{16} \) |
| 5 | \( 1 + 3848885859256 p^{2} T^{2} + \)\(76\!\cdots\!48\)\( p^{4} T^{4} + \)\(97\!\cdots\!84\)\( p^{6} T^{6} + \)\(88\!\cdots\!26\)\( p^{8} T^{8} + \)\(97\!\cdots\!84\)\( p^{44} T^{10} + \)\(76\!\cdots\!48\)\( p^{80} T^{12} + 3848885859256 p^{116} T^{14} + p^{152} T^{16} \) | |
| 11 | \( 1 + \)\(24\!\cdots\!24\)\( T^{2} + \)\(29\!\cdots\!88\)\( T^{4} + \)\(19\!\cdots\!32\)\( p^{2} T^{6} + \)\(10\!\cdots\!74\)\( p^{4} T^{8} + \)\(19\!\cdots\!32\)\( p^{40} T^{10} + \)\(29\!\cdots\!88\)\( p^{76} T^{12} + \)\(24\!\cdots\!24\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 13 | \( ( 1 + 69886183192 T + \)\(52\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!96\)\( p T^{3} + \)\(64\!\cdots\!94\)\( p^{2} T^{4} + \)\(20\!\cdots\!96\)\( p^{20} T^{5} + \)\(52\!\cdots\!00\)\( p^{38} T^{6} + 69886183192 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 17 | \( 1 + \)\(68\!\cdots\!16\)\( T^{2} + \)\(85\!\cdots\!60\)\( p^{2} T^{4} + \)\(56\!\cdots\!36\)\( p^{5} T^{6} + \)\(92\!\cdots\!82\)\( p^{6} T^{8} + \)\(56\!\cdots\!36\)\( p^{43} T^{10} + \)\(85\!\cdots\!60\)\( p^{78} T^{12} + \)\(68\!\cdots\!16\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 19 | \( ( 1 - 1172443419728 T + \)\(50\!\cdots\!52\)\( T^{2} - \)\(49\!\cdots\!96\)\( T^{3} + \)\(14\!\cdots\!34\)\( T^{4} - \)\(49\!\cdots\!96\)\( p^{19} T^{5} + \)\(50\!\cdots\!52\)\( p^{38} T^{6} - 1172443419728 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 23 | \( 1 + \)\(35\!\cdots\!20\)\( T^{2} + \)\(61\!\cdots\!04\)\( T^{4} + \)\(70\!\cdots\!20\)\( T^{6} + \)\(59\!\cdots\!26\)\( T^{8} + \)\(70\!\cdots\!20\)\( p^{38} T^{10} + \)\(61\!\cdots\!04\)\( p^{76} T^{12} + \)\(35\!\cdots\!20\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 29 | \( 1 + \)\(41\!\cdots\!44\)\( T^{2} + \)\(78\!\cdots\!12\)\( T^{4} + \)\(89\!\cdots\!00\)\( T^{6} + \)\(66\!\cdots\!86\)\( T^{8} + \)\(89\!\cdots\!00\)\( p^{38} T^{10} + \)\(78\!\cdots\!12\)\( p^{76} T^{12} + \)\(41\!\cdots\!44\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 31 | \( ( 1 + 230486396113456 T + \)\(43\!\cdots\!60\)\( T^{2} + \)\(58\!\cdots\!04\)\( T^{3} + \)\(82\!\cdots\!54\)\( T^{4} + \)\(58\!\cdots\!04\)\( p^{19} T^{5} + \)\(43\!\cdots\!60\)\( p^{38} T^{6} + 230486396113456 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 750085021566424 T + \)\(15\!\cdots\!08\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!20\)\( p^{19} T^{5} + \)\(15\!\cdots\!08\)\( p^{38} T^{6} + 750085021566424 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 41 | \( 1 + \)\(13\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!92\)\( T^{4} + \)\(51\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!98\)\( T^{8} + \)\(51\!\cdots\!00\)\( p^{38} T^{10} + \)\(10\!\cdots\!92\)\( p^{76} T^{12} + \)\(13\!\cdots\!00\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 43 | \( ( 1 - 620247319381328 T + \)\(18\!\cdots\!20\)\( T^{2} - \)\(65\!\cdots\!32\)\( T^{3} + \)\(15\!\cdots\!86\)\( T^{4} - \)\(65\!\cdots\!32\)\( p^{19} T^{5} + \)\(18\!\cdots\!20\)\( p^{38} T^{6} - 620247319381328 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 47 | \( 1 + \)\(29\!\cdots\!88\)\( T^{2} + \)\(39\!\cdots\!12\)\( T^{4} + \)\(31\!\cdots\!76\)\( T^{6} + \)\(20\!\cdots\!94\)\( T^{8} + \)\(31\!\cdots\!76\)\( p^{38} T^{10} + \)\(39\!\cdots\!12\)\( p^{76} T^{12} + \)\(29\!\cdots\!88\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 53 | \( 1 + \)\(35\!\cdots\!64\)\( T^{2} + \)\(60\!\cdots\!60\)\( T^{4} + \)\(63\!\cdots\!48\)\( T^{6} + \)\(44\!\cdots\!38\)\( T^{8} + \)\(63\!\cdots\!48\)\( p^{38} T^{10} + \)\(60\!\cdots\!60\)\( p^{76} T^{12} + \)\(35\!\cdots\!64\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 59 | \( 1 + \)\(22\!\cdots\!68\)\( T^{2} + \)\(25\!\cdots\!80\)\( T^{4} + \)\(18\!\cdots\!44\)\( T^{6} + \)\(94\!\cdots\!78\)\( T^{8} + \)\(18\!\cdots\!44\)\( p^{38} T^{10} + \)\(25\!\cdots\!80\)\( p^{76} T^{12} + \)\(22\!\cdots\!68\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 61 | \( ( 1 - 56128048864837880 T + \)\(12\!\cdots\!64\)\( T^{2} - \)\(32\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(32\!\cdots\!40\)\( p^{19} T^{5} + \)\(12\!\cdots\!64\)\( p^{38} T^{6} - 56128048864837880 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 75733494970155568 T + \)\(12\!\cdots\!64\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(75\!\cdots\!70\)\( T^{4} + \)\(22\!\cdots\!76\)\( p^{19} T^{5} + \)\(12\!\cdots\!64\)\( p^{38} T^{6} + 75733494970155568 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 71 | \( 1 + \)\(10\!\cdots\!20\)\( T^{2} + \)\(32\!\cdots\!32\)\( T^{4} + \)\(68\!\cdots\!40\)\( T^{6} + \)\(45\!\cdots\!98\)\( T^{8} + \)\(68\!\cdots\!40\)\( p^{38} T^{10} + \)\(32\!\cdots\!32\)\( p^{76} T^{12} + \)\(10\!\cdots\!20\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 73 | \( ( 1 + 368041086891678088 T + \)\(69\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!92\)\( T^{3} + \)\(20\!\cdots\!86\)\( T^{4} + \)\(11\!\cdots\!92\)\( p^{19} T^{5} + \)\(69\!\cdots\!80\)\( p^{38} T^{6} + 368041086891678088 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 934021788005299360 T + \)\(27\!\cdots\!08\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(40\!\cdots\!38\)\( T^{4} + \)\(22\!\cdots\!80\)\( p^{19} T^{5} + \)\(27\!\cdots\!08\)\( p^{38} T^{6} + 934021788005299360 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
| 83 | \( 1 + \)\(80\!\cdots\!88\)\( T^{2} + \)\(53\!\cdots\!72\)\( T^{4} + \)\(21\!\cdots\!76\)\( T^{6} + \)\(76\!\cdots\!34\)\( T^{8} + \)\(21\!\cdots\!76\)\( p^{38} T^{10} + \)\(53\!\cdots\!72\)\( p^{76} T^{12} + \)\(80\!\cdots\!88\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 89 | \( 1 + \)\(42\!\cdots\!20\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{4} + \)\(19\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!18\)\( T^{8} + \)\(19\!\cdots\!40\)\( p^{38} T^{10} + \)\(11\!\cdots\!72\)\( p^{76} T^{12} + \)\(42\!\cdots\!20\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
| 97 | \( ( 1 + 4660331432528518600 T + \)\(21\!\cdots\!24\)\( T^{2} + \)\(73\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!22\)\( T^{4} + \)\(73\!\cdots\!00\)\( p^{19} T^{5} + \)\(21\!\cdots\!24\)\( p^{38} T^{6} + 4660331432528518600 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
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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
−4.60312566703732570543815728376, −4.13473501032801205339500434535, −3.94546724534399899320087289643, −3.94281072831587616626036090798, −3.87298796827807345558123533286, −3.81549079716455875045698192655, −3.77620286225030330611392852168, −3.77068137570807886812356138520, −3.18679243401328595308159335099, −3.05636570892900597705752927996, −2.75271210645783412988743771982, −2.65416410787802129178547103711, −2.63269370613482108116682269949, −2.40766138647587331500333439601, −2.12923923482854740257261139361, −2.02023968817505935184406223652, −2.00731507425467205675425938665, −1.90385385379217404905496078239, −1.73942916613319581832414177339, −1.31740537099559599102058816551, −1.22326310128250278827769758958, −1.13954348960095609799528043317, −1.13200924454983735049420827918, −1.07862529953244484826480902915, −0.920505387714339410446087822147, 0, 0, 0, 0, 0, 0, 0, 0, 0.920505387714339410446087822147, 1.07862529953244484826480902915, 1.13200924454983735049420827918, 1.13954348960095609799528043317, 1.22326310128250278827769758958, 1.31740537099559599102058816551, 1.73942916613319581832414177339, 1.90385385379217404905496078239, 2.00731507425467205675425938665, 2.02023968817505935184406223652, 2.12923923482854740257261139361, 2.40766138647587331500333439601, 2.63269370613482108116682269949, 2.65416410787802129178547103711, 2.75271210645783412988743771982, 3.05636570892900597705752927996, 3.18679243401328595308159335099, 3.77068137570807886812356138520, 3.77620286225030330611392852168, 3.81549079716455875045698192655, 3.87298796827807345558123533286, 3.94281072831587616626036090798, 3.94546724534399899320087289643, 4.13473501032801205339500434535, 4.60312566703732570543815728376
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.