Dirichlet series
| L(s) = 1 | + 2·5-s + 362·7-s + 742·9-s + 902·11-s + 1.55e3·17-s + 6.23e3·19-s − 1.88e4·23-s − 6.85e4·25-s + 724·35-s − 3.35e5·43-s + 1.48e3·45-s − 5.70e5·47-s − 1.80e5·49-s + 1.80e3·55-s − 6.32e5·61-s + 2.68e5·63-s − 8.52e5·73-s + 3.26e5·77-s − 9.72e5·81-s + 4.41e5·83-s + 3.10e3·85-s + 1.24e4·95-s + 6.69e5·99-s − 1.81e6·101-s − 3.76e4·115-s + 5.61e5·119-s − 5.29e6·121-s + ⋯ |
| L(s) = 1 | + 0.0159·5-s + 1.05·7-s + 1.01·9-s + 0.677·11-s + 0.315·17-s + 0.908·19-s − 1.54·23-s − 4.38·25-s + 0.0168·35-s − 4.21·43-s + 0.0162·45-s − 5.49·47-s − 1.53·49-s + 0.0108·55-s − 2.78·61-s + 1.07·63-s − 2.19·73-s + 0.715·77-s − 1.82·81-s + 0.771·83-s + 0.00504·85-s + 0.0145·95-s + 0.689·99-s − 1.76·101-s − 0.0247·115-s + 0.332·119-s − 2.98·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.73269\times 10^{9}\) |
| Root analytic conductor: | \(4.18140\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 19^{8} ,\ ( \ : [3]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(0.1540153675\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1540153675\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 328 p T + 7370512 p T^{2} - 1950884072 p^{2} T^{3} + 70846142174 p^{4} T^{4} - 1950884072 p^{8} T^{5} + 7370512 p^{13} T^{6} - 328 p^{19} T^{7} + p^{24} T^{8} \) | |
| good | 3 | \( 1 - 742 T^{2} + 507523 p T^{4} - 139764998 p^{2} T^{6} + 39023898956 p^{3} T^{8} - 139764998 p^{14} T^{10} + 507523 p^{25} T^{12} - 742 p^{36} T^{14} + p^{48} T^{16} \) |
| 5 | \( ( 1 - T + 6858 p T^{2} + 57133 p^{2} T^{3} + 4953194 p^{3} T^{4} + 57133 p^{8} T^{5} + 6858 p^{13} T^{6} - p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 7 | \( ( 1 - 181 T + 139633 T^{2} + 29351086 T^{3} - 320349574 T^{4} + 29351086 p^{6} T^{5} + 139633 p^{12} T^{6} - 181 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 11 | \( ( 1 - 41 p T + 2951358 T^{2} - 3849343493 T^{3} + 4110119555458 T^{4} - 3849343493 p^{6} T^{5} + 2951358 p^{12} T^{6} - 41 p^{19} T^{7} + p^{24} T^{8} )^{2} \) | |
| 13 | \( 1 - 18101774 T^{2} + 149249945971081 T^{4} - \)\(86\!\cdots\!26\)\( T^{6} + \)\(43\!\cdots\!16\)\( T^{8} - \)\(86\!\cdots\!26\)\( p^{12} T^{10} + 149249945971081 p^{24} T^{12} - 18101774 p^{36} T^{14} + p^{48} T^{16} \) | |
| 17 | \( ( 1 - 775 T + 76306401 T^{2} - 62323644578 T^{3} + 2574852969478894 T^{4} - 62323644578 p^{6} T^{5} + 76306401 p^{12} T^{6} - 775 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 23 | \( ( 1 + 9410 T + 537866745 T^{2} + 3615134698390 T^{3} + 115088224709036668 T^{4} + 3615134698390 p^{6} T^{5} + 537866745 p^{12} T^{6} + 9410 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 29 | \( 1 - 1955972438 T^{2} + 1491101576657216569 T^{4} - \)\(39\!\cdots\!02\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{8} - \)\(39\!\cdots\!02\)\( p^{12} T^{10} + 1491101576657216569 p^{24} T^{12} - 1955972438 p^{36} T^{14} + p^{48} T^{16} \) | |
| 31 | \( 1 - 1360240952 T^{2} + 2738159624622996748 T^{4} - \)\(21\!\cdots\!68\)\( T^{6} + \)\(27\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!68\)\( p^{12} T^{10} + 2738159624622996748 p^{24} T^{12} - 1360240952 p^{36} T^{14} + p^{48} T^{16} \) | |
| 37 | \( 1 - 7244329016 T^{2} + 23581771984582865740 T^{4} - \)\(37\!\cdots\!88\)\( T^{6} + \)\(48\!\cdots\!10\)\( T^{8} - \)\(37\!\cdots\!88\)\( p^{12} T^{10} + 23581771984582865740 p^{24} T^{12} - 7244329016 p^{36} T^{14} + p^{48} T^{16} \) | |
| 41 | \( 1 - 4812538424 T^{2} + 68132340249279236236 T^{4} - \)\(21\!\cdots\!16\)\( T^{6} + \)\(48\!\cdots\!26\)\( p T^{8} - \)\(21\!\cdots\!16\)\( p^{12} T^{10} + 68132340249279236236 p^{24} T^{12} - 4812538424 p^{36} T^{14} + p^{48} T^{16} \) | |
| 43 | \( ( 1 + 167521 T + 28059854686 T^{2} + 2860511677640443 T^{3} + \)\(27\!\cdots\!70\)\( T^{4} + 2860511677640443 p^{6} T^{5} + 28059854686 p^{12} T^{6} + 167521 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 47 | \( ( 1 + 285197 T + 53014565646 T^{2} + 6790341576595363 T^{3} + \)\(75\!\cdots\!74\)\( T^{4} + 6790341576595363 p^{6} T^{5} + 53014565646 p^{12} T^{6} + 285197 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 53 | \( 1 - 106845552662 T^{2} + \)\(57\!\cdots\!21\)\( T^{4} - \)\(20\!\cdots\!82\)\( T^{6} + \)\(53\!\cdots\!08\)\( T^{8} - \)\(20\!\cdots\!82\)\( p^{12} T^{10} + \)\(57\!\cdots\!21\)\( p^{24} T^{12} - 106845552662 p^{36} T^{14} + p^{48} T^{16} \) | |
| 59 | \( 1 - 283261383806 T^{2} + \)\(36\!\cdots\!69\)\( T^{4} - \)\(28\!\cdots\!06\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{8} - \)\(28\!\cdots\!06\)\( p^{12} T^{10} + \)\(36\!\cdots\!69\)\( p^{24} T^{12} - 283261383806 p^{36} T^{14} + p^{48} T^{16} \) | |
| 61 | \( ( 1 + 316007 T + 182153576986 T^{2} + 45019267728697357 T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + 45019267728697357 p^{6} T^{5} + 182153576986 p^{12} T^{6} + 316007 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 67 | \( 1 - 284119902326 T^{2} + \)\(40\!\cdots\!01\)\( T^{4} - \)\(44\!\cdots\!94\)\( T^{6} + \)\(42\!\cdots\!76\)\( T^{8} - \)\(44\!\cdots\!94\)\( p^{12} T^{10} + \)\(40\!\cdots\!01\)\( p^{24} T^{12} - 284119902326 p^{36} T^{14} + p^{48} T^{16} \) | |
| 71 | \( 1 - 550447281824 T^{2} + \)\(16\!\cdots\!32\)\( T^{4} - \)\(34\!\cdots\!36\)\( T^{6} + \)\(52\!\cdots\!54\)\( T^{8} - \)\(34\!\cdots\!36\)\( p^{12} T^{10} + \)\(16\!\cdots\!32\)\( p^{24} T^{12} - 550447281824 p^{36} T^{14} + p^{48} T^{16} \) | |
| 73 | \( ( 1 + 426469 T + 412154349109 T^{2} + 175844843253819082 T^{3} + \)\(80\!\cdots\!22\)\( T^{4} + 175844843253819082 p^{6} T^{5} + 412154349109 p^{12} T^{6} + 426469 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 79 | \( 1 - 935745384632 T^{2} + \)\(48\!\cdots\!64\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{6} + \)\(48\!\cdots\!78\)\( T^{8} - \)\(17\!\cdots\!52\)\( p^{12} T^{10} + \)\(48\!\cdots\!64\)\( p^{24} T^{12} - 935745384632 p^{36} T^{14} + p^{48} T^{16} \) | |
| 83 | \( ( 1 - 220600 T + 1020667843728 T^{2} - 182031252335003816 T^{3} + \)\(45\!\cdots\!30\)\( T^{4} - 182031252335003816 p^{6} T^{5} + 1020667843728 p^{12} T^{6} - 220600 p^{18} T^{7} + p^{24} T^{8} )^{2} \) | |
| 89 | \( 1 - 1390950826016 T^{2} + \)\(76\!\cdots\!16\)\( T^{4} - \)\(73\!\cdots\!44\)\( T^{6} - \)\(68\!\cdots\!94\)\( T^{8} - \)\(73\!\cdots\!44\)\( p^{12} T^{10} + \)\(76\!\cdots\!16\)\( p^{24} T^{12} - 1390950826016 p^{36} T^{14} + p^{48} T^{16} \) | |
| 97 | \( 1 - 5125637527808 T^{2} + \)\(12\!\cdots\!48\)\( T^{4} - \)\(18\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!02\)\( T^{8} - \)\(18\!\cdots\!72\)\( p^{12} T^{10} + \)\(12\!\cdots\!48\)\( p^{24} T^{12} - 5125637527808 p^{36} T^{14} + p^{48} T^{16} \) | |
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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
−5.51971255905067304947644853405, −5.22457789001735216481205751997, −5.08627815986691194357724860715, −4.90187647161800223428812063302, −4.87917508834811940726653087756, −4.49790604896557482901876417388, −4.33644624358380704148085202061, −4.18133599241282527426623013226, −4.02679181464871753979399686322, −3.76795837058055505795475155777, −3.63653121635304310063833698014, −3.19618052833142988101970536198, −3.15344402168769920497561295260, −3.14135421653420517638554082514, −2.79568752101768895004325272647, −2.30072435624183687155066593109, −1.90481307981278954736588639204, −1.76492581275011216445786466243, −1.71789443610224450913029908938, −1.52203592176866892054030876066, −1.38481190006585617389101392727, −1.37457836824521052146153700663, −0.56813098159546206864079020673, −0.13847219126654255623326052969, −0.099918861017052233589202121178, 0.099918861017052233589202121178, 0.13847219126654255623326052969, 0.56813098159546206864079020673, 1.37457836824521052146153700663, 1.38481190006585617389101392727, 1.52203592176866892054030876066, 1.71789443610224450913029908938, 1.76492581275011216445786466243, 1.90481307981278954736588639204, 2.30072435624183687155066593109, 2.79568752101768895004325272647, 3.14135421653420517638554082514, 3.15344402168769920497561295260, 3.19618052833142988101970536198, 3.63653121635304310063833698014, 3.76795837058055505795475155777, 4.02679181464871753979399686322, 4.18133599241282527426623013226, 4.33644624358380704148085202061, 4.49790604896557482901876417388, 4.87917508834811940726653087756, 4.90187647161800223428812063302, 5.08627815986691194357724860715, 5.22457789001735216481205751997, 5.51971255905067304947644853405
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.