Dirichlet series
| L(s) = 1 | + 2.36e7·5-s − 4.64e9·9-s + 1.39e11·13-s − 5.33e12·17-s + 8.27e13·25-s + 4.30e14·29-s − 4.61e15·37-s − 2.47e16·41-s − 1.09e17·45-s + 1.44e17·49-s − 5.39e17·53-s − 2.65e17·61-s + 3.30e18·65-s − 1.61e18·73-s + 1.35e19·81-s − 1.26e20·85-s + 1.22e20·89-s + 1.10e19·97-s + 8.66e19·101-s + 3.94e20·109-s + 6.95e20·113-s − 6.49e20·117-s + 4.66e21·121-s − 6.74e20·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.41·5-s − 4/3·9-s + 1.01·13-s − 2.64·17-s + 0.868·25-s + 1.02·29-s − 0.960·37-s − 1.84·41-s − 3.22·45-s + 1.81·49-s − 3.08·53-s − 0.372·61-s + 2.45·65-s − 0.375·73-s + 10/9·81-s − 6.40·85-s + 3.93·89-s + 0.149·97-s + 0.784·101-s + 1.66·109-s + 2.04·113-s − 1.35·117-s + 6.93·121-s − 0.724·125-s + 2.47·145-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(21-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+10)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.80775\times 10^{16}\) |
| Root analytic conductor: | \(11.0311\) |
| Motivic weight: | \(20\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [10]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{21}{2})\) | \(\approx\) | \(31.47578632\) |
| \(L(\frac12)\) | \(\approx\) | \(31.47578632\) |
| \(L(11)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{19} T^{2} )^{4} \) | |
| good | 5 | \( ( 1 - 2362968 p T + 268781064604 p^{4} T^{2} - 3709352112775842312 p^{4} T^{3} + \)\(15\!\cdots\!54\)\( p^{6} T^{4} - 3709352112775842312 p^{24} T^{5} + 268781064604 p^{44} T^{6} - 2362968 p^{61} T^{7} + p^{80} T^{8} )^{2} \) |
| 7 | \( 1 - 144616868639342408 T^{2} + \)\(36\!\cdots\!40\)\( p T^{4} - \)\(47\!\cdots\!16\)\( p^{2} T^{6} + \)\(20\!\cdots\!42\)\( p^{6} T^{8} - \)\(47\!\cdots\!16\)\( p^{42} T^{10} + \)\(36\!\cdots\!40\)\( p^{81} T^{12} - 144616868639342408 p^{120} T^{14} + p^{160} T^{16} \) | |
| 11 | \( 1 - \)\(46\!\cdots\!40\)\( T^{2} + \)\(99\!\cdots\!24\)\( T^{4} - \)\(94\!\cdots\!20\)\( p^{3} T^{6} + \)\(70\!\cdots\!06\)\( p^{4} T^{8} - \)\(94\!\cdots\!20\)\( p^{43} T^{10} + \)\(99\!\cdots\!24\)\( p^{80} T^{12} - \)\(46\!\cdots\!40\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 13 | \( ( 1 - 69851967208 T - \)\(55\!\cdots\!92\)\( T^{2} - \)\(10\!\cdots\!52\)\( p T^{3} + \)\(39\!\cdots\!50\)\( p^{2} T^{4} - \)\(10\!\cdots\!52\)\( p^{21} T^{5} - \)\(55\!\cdots\!92\)\( p^{40} T^{6} - 69851967208 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 17 | \( ( 1 + 2667523843224 T + \)\(67\!\cdots\!16\)\( p T^{2} + \)\(10\!\cdots\!20\)\( p^{2} T^{3} + \)\(13\!\cdots\!42\)\( p^{3} T^{4} + \)\(10\!\cdots\!20\)\( p^{22} T^{5} + \)\(67\!\cdots\!16\)\( p^{41} T^{6} + 2667523843224 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 19 | \( 1 - \)\(10\!\cdots\!48\)\( T^{2} + \)\(18\!\cdots\!80\)\( p^{2} T^{4} - \)\(25\!\cdots\!84\)\( p^{4} T^{6} + \)\(30\!\cdots\!98\)\( p^{6} T^{8} - \)\(25\!\cdots\!84\)\( p^{44} T^{10} + \)\(18\!\cdots\!80\)\( p^{82} T^{12} - \)\(10\!\cdots\!48\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 23 | \( 1 - \)\(59\!\cdots\!64\)\( T^{2} + \)\(20\!\cdots\!68\)\( T^{4} - \)\(48\!\cdots\!12\)\( T^{6} + \)\(92\!\cdots\!14\)\( T^{8} - \)\(48\!\cdots\!12\)\( p^{40} T^{10} + \)\(20\!\cdots\!68\)\( p^{80} T^{12} - \)\(59\!\cdots\!64\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 29 | \( ( 1 - 215340717323736 T + \)\(13\!\cdots\!48\)\( T^{2} + \)\(15\!\cdots\!72\)\( T^{3} - \)\(70\!\cdots\!06\)\( T^{4} + \)\(15\!\cdots\!72\)\( p^{20} T^{5} + \)\(13\!\cdots\!48\)\( p^{40} T^{6} - 215340717323736 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 31 | \( 1 - \)\(51\!\cdots\!60\)\( p T^{2} + \)\(12\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{6} - \)\(89\!\cdots\!94\)\( p^{2} T^{8} - \)\(19\!\cdots\!80\)\( p^{40} T^{10} + \)\(12\!\cdots\!84\)\( p^{80} T^{12} - \)\(51\!\cdots\!60\)\( p^{121} T^{14} + p^{160} T^{16} \) | |
| 37 | \( ( 1 + 2309957346165304 T + \)\(50\!\cdots\!08\)\( T^{2} + \)\(19\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!34\)\( T^{4} + \)\(19\!\cdots\!72\)\( p^{20} T^{5} + \)\(50\!\cdots\!08\)\( p^{40} T^{6} + 2309957346165304 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 41 | \( ( 1 + 12367250275117464 T + \)\(52\!\cdots\!72\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} + \)\(36\!\cdots\!00\)\( p^{20} T^{5} + \)\(52\!\cdots\!72\)\( p^{40} T^{6} + 12367250275117464 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 43 | \( 1 - \)\(20\!\cdots\!88\)\( T^{2} + \)\(19\!\cdots\!80\)\( T^{4} - \)\(12\!\cdots\!04\)\( T^{6} + \)\(62\!\cdots\!78\)\( T^{8} - \)\(12\!\cdots\!04\)\( p^{40} T^{10} + \)\(19\!\cdots\!80\)\( p^{80} T^{12} - \)\(20\!\cdots\!88\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 47 | \( 1 - \)\(13\!\cdots\!88\)\( T^{2} + \)\(91\!\cdots\!20\)\( T^{4} - \)\(42\!\cdots\!04\)\( T^{6} + \)\(13\!\cdots\!38\)\( T^{8} - \)\(42\!\cdots\!04\)\( p^{40} T^{10} + \)\(91\!\cdots\!20\)\( p^{80} T^{12} - \)\(13\!\cdots\!88\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 53 | \( ( 1 + 269857729481267880 T + \)\(13\!\cdots\!32\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!98\)\( T^{4} + \)\(23\!\cdots\!00\)\( p^{20} T^{5} + \)\(13\!\cdots\!32\)\( p^{40} T^{6} + 269857729481267880 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 59 | \( 1 - \)\(63\!\cdots\!60\)\( T^{2} + \)\(23\!\cdots\!24\)\( T^{4} - \)\(84\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!46\)\( T^{8} - \)\(84\!\cdots\!80\)\( p^{40} T^{10} + \)\(23\!\cdots\!24\)\( p^{80} T^{12} - \)\(63\!\cdots\!60\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 61 | \( ( 1 + 132768598477764088 T + \)\(13\!\cdots\!80\)\( T^{2} - \)\(74\!\cdots\!96\)\( T^{3} + \)\(82\!\cdots\!58\)\( T^{4} - \)\(74\!\cdots\!96\)\( p^{20} T^{5} + \)\(13\!\cdots\!80\)\( p^{40} T^{6} + 132768598477764088 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 67 | \( 1 - \)\(16\!\cdots\!88\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{4} - \)\(60\!\cdots\!04\)\( T^{6} + \)\(22\!\cdots\!18\)\( T^{8} - \)\(60\!\cdots\!04\)\( p^{40} T^{10} + \)\(12\!\cdots\!20\)\( p^{80} T^{12} - \)\(16\!\cdots\!88\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 71 | \( 1 - \)\(62\!\cdots\!08\)\( T^{2} + \)\(18\!\cdots\!20\)\( T^{4} - \)\(33\!\cdots\!04\)\( T^{6} + \)\(43\!\cdots\!38\)\( T^{8} - \)\(33\!\cdots\!04\)\( p^{40} T^{10} + \)\(18\!\cdots\!20\)\( p^{80} T^{12} - \)\(62\!\cdots\!08\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 73 | \( ( 1 + 807458747610702200 T + \)\(49\!\cdots\!92\)\( T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!78\)\( T^{4} + \)\(39\!\cdots\!60\)\( p^{20} T^{5} + \)\(49\!\cdots\!92\)\( p^{40} T^{6} + 807458747610702200 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 79 | \( 1 - \)\(55\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(22\!\cdots\!32\)\( T^{6} + \)\(23\!\cdots\!14\)\( T^{8} - \)\(22\!\cdots\!32\)\( p^{40} T^{10} + \)\(14\!\cdots\!56\)\( p^{80} T^{12} - \)\(55\!\cdots\!32\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 83 | \( 1 - \)\(12\!\cdots\!36\)\( T^{2} + \)\(89\!\cdots\!20\)\( p T^{4} - \)\(28\!\cdots\!84\)\( T^{6} + \)\(80\!\cdots\!74\)\( T^{8} - \)\(28\!\cdots\!84\)\( p^{40} T^{10} + \)\(89\!\cdots\!20\)\( p^{81} T^{12} - \)\(12\!\cdots\!36\)\( p^{120} T^{14} + p^{160} T^{16} \) | |
| 89 | \( ( 1 - 61397064551112916104 T + \)\(43\!\cdots\!40\)\( T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(64\!\cdots\!34\)\( T^{4} - \)\(17\!\cdots\!76\)\( p^{20} T^{5} + \)\(43\!\cdots\!40\)\( p^{40} T^{6} - 61397064551112916104 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
| 97 | \( ( 1 - 5503776850440636040 T + \)\(10\!\cdots\!16\)\( T^{2} - \)\(67\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!06\)\( T^{4} - \)\(67\!\cdots\!80\)\( p^{20} T^{5} + \)\(10\!\cdots\!16\)\( p^{40} T^{6} - 5503776850440636040 p^{60} T^{7} + p^{80} T^{8} )^{2} \) | |
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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
−4.24104543134250515215586213290, −3.54113477376850264289966624365, −3.44861203597719181598410635367, −3.40624876297677406576389789233, −3.25944627312758567977116135527, −3.24948088557875003992617201750, −3.09490082963954822018295083560, −2.81881398221205561100858977892, −2.80708747984296831793936917889, −2.23313108656234997353143427631, −2.20234367413530936654891330422, −2.06100884109745693013339109922, −1.94272675276518223122039836898, −1.92940664354369792872079005800, −1.85253409038165211780222852228, −1.83345659650510058077432750228, −1.68015107989761118823608244637, −1.18518047552740799033202499876, −1.01800450656903016829673839550, −0.831565067506123662888417686900, −0.66857881045732806646290904467, −0.63101512869281337688003093323, −0.42481461019137211268688896206, −0.31834891933984249596416357708, −0.23278917407446541868925250770, 0.23278917407446541868925250770, 0.31834891933984249596416357708, 0.42481461019137211268688896206, 0.63101512869281337688003093323, 0.66857881045732806646290904467, 0.831565067506123662888417686900, 1.01800450656903016829673839550, 1.18518047552740799033202499876, 1.68015107989761118823608244637, 1.83345659650510058077432750228, 1.85253409038165211780222852228, 1.92940664354369792872079005800, 1.94272675276518223122039836898, 2.06100884109745693013339109922, 2.20234367413530936654891330422, 2.23313108656234997353143427631, 2.80708747984296831793936917889, 2.81881398221205561100858977892, 3.09490082963954822018295083560, 3.24948088557875003992617201750, 3.25944627312758567977116135527, 3.40624876297677406576389789233, 3.44861203597719181598410635367, 3.54113477376850264289966624365, 4.24104543134250515215586213290
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.