Dirichlet series
| L(s) = 1 | + 1.36e4·7-s − 2.62e4·9-s + 8.30e3·17-s + 4.61e6·23-s + 5.43e6·25-s + 7.49e6·31-s − 4.35e7·41-s − 4.93e7·47-s − 1.05e8·49-s − 3.57e8·63-s + 7.41e8·71-s − 1.50e9·73-s − 1.00e9·79-s + 4.30e8·81-s − 1.33e9·89-s − 9.04e8·97-s − 7.92e8·103-s − 8.61e9·113-s + 1.13e8·119-s + 6.88e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.17e8·153-s + ⋯ |
| L(s) = 1 | + 2.14·7-s − 4/3·9-s + 0.0241·17-s + 3.43·23-s + 2.78·25-s + 1.45·31-s − 2.40·41-s − 1.47·47-s − 2.61·49-s − 2.86·63-s + 3.46·71-s − 6.21·73-s − 2.91·79-s + 10/9·81-s − 2.25·89-s − 1.03·97-s − 0.693·103-s − 4.96·113-s + 0.0517·119-s + 2.91·121-s − 0.0321·153-s + 7.37·161-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{56} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.34071\times 10^{18}\) |
| Root analytic conductor: | \(14.0632\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(0.1855532101\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1855532101\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{8} T^{2} )^{4} \) | |
| good | 5 | \( 1 - 5435048 T^{2} + 20893425650876 T^{4} - 2321812585418275992 p^{2} T^{6} + \)\(20\!\cdots\!54\)\( p^{4} T^{8} - 2321812585418275992 p^{20} T^{10} + 20893425650876 p^{36} T^{12} - 5435048 p^{54} T^{14} + p^{72} T^{16} \) |
| 7 | \( ( 1 - 6816 T + 17494692 p T^{2} - 11388268320 p^{2} T^{3} + 18913377869578 p^{3} T^{4} - 11388268320 p^{11} T^{5} + 17494692 p^{19} T^{6} - 6816 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 11 | \( 1 - 6884938520 T^{2} + 2965172912109472948 p T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(29\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!16\)\( p^{18} T^{10} + 2965172912109472948 p^{37} T^{12} - 6884938520 p^{54} T^{14} + p^{72} T^{16} \) | |
| 13 | \( 1 - 57037171240 T^{2} + \)\(14\!\cdots\!60\)\( T^{4} - \)\(23\!\cdots\!76\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} - \)\(23\!\cdots\!76\)\( p^{18} T^{10} + \)\(14\!\cdots\!60\)\( p^{36} T^{12} - 57037171240 p^{54} T^{14} + p^{72} T^{16} \) | |
| 17 | \( ( 1 - 4152 T + 263140496604 T^{2} + 47861357636656632 T^{3} + \)\(31\!\cdots\!78\)\( T^{4} + 47861357636656632 p^{9} T^{5} + 263140496604 p^{18} T^{6} - 4152 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 19 | \( 1 - 741411893592 T^{2} + \)\(38\!\cdots\!28\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!42\)\( T^{8} - \)\(11\!\cdots\!28\)\( p^{18} T^{10} + \)\(38\!\cdots\!28\)\( p^{36} T^{12} - 741411893592 p^{54} T^{14} + p^{72} T^{16} \) | |
| 23 | \( ( 1 - 2306304 T + 8281005731548 T^{2} - 12314633297082856704 T^{3} + \)\(23\!\cdots\!90\)\( T^{4} - 12314633297082856704 p^{9} T^{5} + 8281005731548 p^{18} T^{6} - 2306304 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 29 | \( 1 + 10389690701976 T^{2} + \)\(14\!\cdots\!56\)\( p T^{4} - \)\(56\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!38\)\( T^{8} - \)\(56\!\cdots\!80\)\( p^{18} T^{10} + \)\(14\!\cdots\!56\)\( p^{37} T^{12} + 10389690701976 p^{54} T^{14} + p^{72} T^{16} \) | |
| 31 | \( ( 1 - 3749664 T + 100674806735580 T^{2} - \)\(27\!\cdots\!44\)\( T^{3} + \)\(39\!\cdots\!02\)\( T^{4} - \)\(27\!\cdots\!44\)\( p^{9} T^{5} + 100674806735580 p^{18} T^{6} - 3749664 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 37 | \( 1 - 550539626288616 T^{2} + \)\(10\!\cdots\!92\)\( T^{4} - \)\(16\!\cdots\!12\)\( p T^{6} - \)\(15\!\cdots\!86\)\( T^{8} - \)\(16\!\cdots\!12\)\( p^{19} T^{10} + \)\(10\!\cdots\!92\)\( p^{36} T^{12} - 550539626288616 p^{54} T^{14} + p^{72} T^{16} \) | |
| 41 | \( ( 1 + 21759448 T + 779819408639484 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!06\)\( T^{4} + \)\(14\!\cdots\!80\)\( p^{9} T^{5} + 779819408639484 p^{18} T^{6} + 21759448 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 43 | \( 1 - 2639304093984792 T^{2} + \)\(33\!\cdots\!04\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!50\)\( T^{8} - \)\(28\!\cdots\!00\)\( p^{18} T^{10} + \)\(33\!\cdots\!04\)\( p^{36} T^{12} - 2639304093984792 p^{54} T^{14} + p^{72} T^{16} \) | |
| 47 | \( ( 1 + 24691008 T + 2410036675923004 T^{2} + \)\(20\!\cdots\!32\)\( T^{3} + \)\(28\!\cdots\!78\)\( T^{4} + \)\(20\!\cdots\!32\)\( p^{9} T^{5} + 2410036675923004 p^{18} T^{6} + 24691008 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 53 | \( 1 - 2620853843021352 T^{2} + \)\(20\!\cdots\!76\)\( T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(33\!\cdots\!86\)\( T^{8} - \)\(54\!\cdots\!84\)\( p^{18} T^{10} + \)\(20\!\cdots\!76\)\( p^{36} T^{12} - 2620853843021352 p^{54} T^{14} + p^{72} T^{16} \) | |
| 59 | \( 1 - 47982788636552344 T^{2} + \)\(11\!\cdots\!84\)\( T^{4} - \)\(17\!\cdots\!76\)\( T^{6} + \)\(17\!\cdots\!90\)\( T^{8} - \)\(17\!\cdots\!76\)\( p^{18} T^{10} + \)\(11\!\cdots\!84\)\( p^{36} T^{12} - 47982788636552344 p^{54} T^{14} + p^{72} T^{16} \) | |
| 61 | \( 1 - 86131957542763688 T^{2} + \)\(33\!\cdots\!56\)\( T^{4} - \)\(75\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!54\)\( T^{8} - \)\(75\!\cdots\!08\)\( p^{18} T^{10} + \)\(33\!\cdots\!56\)\( p^{36} T^{12} - 86131957542763688 p^{54} T^{14} + p^{72} T^{16} \) | |
| 67 | \( 1 - 43639592995907288 T^{2} + \)\(22\!\cdots\!64\)\( T^{4} - \)\(60\!\cdots\!60\)\( T^{6} + \)\(20\!\cdots\!90\)\( T^{8} - \)\(60\!\cdots\!60\)\( p^{18} T^{10} + \)\(22\!\cdots\!64\)\( p^{36} T^{12} - 43639592995907288 p^{54} T^{14} + p^{72} T^{16} \) | |
| 71 | \( ( 1 - 370875648 T + 140809684178445724 T^{2} - \)\(25\!\cdots\!32\)\( T^{3} + \)\(65\!\cdots\!50\)\( T^{4} - \)\(25\!\cdots\!32\)\( p^{9} T^{5} + 140809684178445724 p^{18} T^{6} - 370875648 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 753951720 T + 382588289020544764 T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!26\)\( T^{4} + \)\(13\!\cdots\!20\)\( p^{9} T^{5} + 382588289020544764 p^{18} T^{6} + 753951720 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 504186720 T + 287669282447248284 T^{2} + \)\(88\!\cdots\!88\)\( T^{3} + \)\(41\!\cdots\!78\)\( T^{4} + \)\(88\!\cdots\!88\)\( p^{9} T^{5} + 287669282447248284 p^{18} T^{6} + 504186720 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 83 | \( 1 - 941769227507077976 T^{2} + \)\(43\!\cdots\!56\)\( T^{4} - \)\(13\!\cdots\!36\)\( T^{6} + \)\(28\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!36\)\( p^{18} T^{10} + \)\(43\!\cdots\!56\)\( p^{36} T^{12} - 941769227507077976 p^{54} T^{14} + p^{72} T^{16} \) | |
| 89 | \( ( 1 + 668517224 T + 916953720151884348 T^{2} + \)\(59\!\cdots\!72\)\( T^{3} + \)\(41\!\cdots\!62\)\( T^{4} + \)\(59\!\cdots\!72\)\( p^{9} T^{5} + 916953720151884348 p^{18} T^{6} + 668517224 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 97 | \( ( 1 + 452408968 T + 2361010511996232860 T^{2} + \)\(92\!\cdots\!04\)\( T^{3} + \)\(25\!\cdots\!82\)\( T^{4} + \)\(92\!\cdots\!04\)\( p^{9} T^{5} + 2361010511996232860 p^{18} T^{6} + 452408968 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
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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
−3.46500562820703275783027174727, −3.34849318874416848711015088993, −3.19371180620882392297150789291, −3.19364735505993219170434657755, −2.99271533915265434366917361266, −2.81277171185032955521382930704, −2.77099000454211224983123177451, −2.56520585105877906452247326422, −2.39329298789251291737317427228, −2.28495495700837836501449452144, −2.23501794170738129759464609022, −2.02241124925216966422200342960, −1.73037298776615854183797815340, −1.55738163827896759980346924457, −1.43936647983966244321785634031, −1.23457061604569464963924878087, −1.20015723519305237350328197586, −1.17827954450295474573741768169, −1.10001001593556781052250639235, −1.04291844149129832418222866307, −0.978508022618680261705702167875, −0.33899419126839113949954568720, −0.23117516619508811698304751487, −0.094361886474559750799705281455, −0.079624986441764913443266261091, 0.079624986441764913443266261091, 0.094361886474559750799705281455, 0.23117516619508811698304751487, 0.33899419126839113949954568720, 0.978508022618680261705702167875, 1.04291844149129832418222866307, 1.10001001593556781052250639235, 1.17827954450295474573741768169, 1.20015723519305237350328197586, 1.23457061604569464963924878087, 1.43936647983966244321785634031, 1.55738163827896759980346924457, 1.73037298776615854183797815340, 2.02241124925216966422200342960, 2.23501794170738129759464609022, 2.28495495700837836501449452144, 2.39329298789251291737317427228, 2.56520585105877906452247326422, 2.77099000454211224983123177451, 2.81277171185032955521382930704, 2.99271533915265434366917361266, 3.19364735505993219170434657755, 3.19371180620882392297150789291, 3.34849318874416848711015088993, 3.46500562820703275783027174727
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.