Dirichlet series
| L(s) = 1 | − 4.80e3·7-s + 5.90e4·9-s − 1.02e5·17-s − 3.41e6·23-s + 6.60e6·25-s − 8.03e5·31-s − 2.18e6·41-s − 7.43e6·47-s − 1.37e8·49-s − 2.83e8·63-s − 5.60e8·71-s − 5.23e8·73-s + 2.48e8·79-s + 1.60e9·81-s + 7.44e8·89-s − 9.93e6·97-s − 2.78e9·103-s + 7.60e8·113-s + 4.89e8·119-s + 1.06e10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6.02e9·153-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2.99·9-s − 0.296·17-s − 2.54·23-s + 3.37·25-s − 0.156·31-s − 0.120·41-s − 0.222·47-s − 3.41·49-s − 2.26·63-s − 2.61·71-s − 2.15·73-s + 0.719·79-s + 4.14·81-s + 1.25·89-s − 0.0113·97-s − 2.43·103-s + 0.438·113-s + 0.223·119-s + 4.50·121-s − 0.888·153-s + 1.92·161-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.44376\times 10^{9}\) |
| Root analytic conductor: | \(4.05969\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{40} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(4.512697079\) |
| \(L(\frac12)\) | \(\approx\) | \(4.512697079\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 59048 T^{2} + 69594932 p^{3} T^{4} - 6090909784 p^{8} T^{6} + 1071668796386710 p^{6} T^{8} - 6090909784 p^{26} T^{10} + 69594932 p^{39} T^{12} - 59048 p^{54} T^{14} + p^{72} T^{16} \) |
| 5 | \( 1 - 6600808 T^{2} + 17461449664316 T^{4} - 969001730842818264 p^{2} T^{6} + \)\(49\!\cdots\!78\)\( p^{4} T^{8} - 969001730842818264 p^{20} T^{10} + 17461449664316 p^{36} T^{12} - 6600808 p^{54} T^{14} + p^{72} T^{16} \) | |
| 7 | \( ( 1 + 2400 T + 11068292 p T^{2} - 255830496 p^{3} T^{3} + 8426386960458 p^{3} T^{4} - 255830496 p^{12} T^{5} + 11068292 p^{19} T^{6} + 2400 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 11 | \( 1 - 10626901608 T^{2} + 58781092281363321020 T^{4} - \)\(22\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!38\)\( T^{8} - \)\(22\!\cdots\!52\)\( p^{18} T^{10} + 58781092281363321020 p^{36} T^{12} - 10626901608 p^{54} T^{14} + p^{72} T^{16} \) | |
| 13 | \( 1 - 52177437864 T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(19\!\cdots\!44\)\( T^{6} + \)\(22\!\cdots\!38\)\( T^{8} - \)\(19\!\cdots\!44\)\( p^{18} T^{10} + \)\(12\!\cdots\!60\)\( p^{36} T^{12} - 52177437864 p^{54} T^{14} + p^{72} T^{16} \) | |
| 17 | \( ( 1 + 3000 p T + 277845292252 T^{2} + 28596206694990600 T^{3} + \)\(14\!\cdots\!62\)\( p^{2} T^{4} + 28596206694990600 p^{9} T^{5} + 277845292252 p^{18} T^{6} + 3000 p^{28} T^{7} + p^{36} T^{8} )^{2} \) | |
| 19 | \( 1 - 1485325196328 T^{2} + \)\(12\!\cdots\!88\)\( T^{4} - \)\(65\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!58\)\( T^{8} - \)\(65\!\cdots\!60\)\( p^{18} T^{10} + \)\(12\!\cdots\!88\)\( p^{36} T^{12} - 1485325196328 p^{54} T^{14} + p^{72} T^{16} \) | |
| 23 | \( ( 1 + 1706016 T + 3178422376156 T^{2} + 5879850573514848 p T^{3} + \)\(67\!\cdots\!42\)\( T^{4} + 5879850573514848 p^{10} T^{5} + 3178422376156 p^{18} T^{6} + 1706016 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 29 | \( 1 - 78392306727720 T^{2} + \)\(28\!\cdots\!40\)\( T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!38\)\( T^{8} - \)\(67\!\cdots\!00\)\( p^{18} T^{10} + \)\(28\!\cdots\!40\)\( p^{36} T^{12} - 78392306727720 p^{54} T^{14} + p^{72} T^{16} \) | |
| 31 | \( ( 1 + 401792 T + 86180537957500 T^{2} + 37247601810974485376 T^{3} + \)\(32\!\cdots\!74\)\( T^{4} + 37247601810974485376 p^{9} T^{5} + 86180537957500 p^{18} T^{6} + 401792 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 37 | \( 1 - 713593321977192 T^{2} + \)\(24\!\cdots\!96\)\( T^{4} - \)\(53\!\cdots\!04\)\( T^{6} + \)\(82\!\cdots\!86\)\( T^{8} - \)\(53\!\cdots\!04\)\( p^{18} T^{10} + \)\(24\!\cdots\!96\)\( p^{36} T^{12} - 713593321977192 p^{54} T^{14} + p^{72} T^{16} \) | |
| 41 | \( ( 1 + 1090392 T + 1059071398836988 T^{2} + \)\(88\!\cdots\!32\)\( T^{3} + \)\(48\!\cdots\!74\)\( T^{4} + \)\(88\!\cdots\!32\)\( p^{9} T^{5} + 1059071398836988 p^{18} T^{6} + 1090392 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 43 | \( 1 - 1649449028392296 T^{2} + \)\(12\!\cdots\!08\)\( T^{4} - \)\(66\!\cdots\!36\)\( T^{6} + \)\(34\!\cdots\!62\)\( T^{8} - \)\(66\!\cdots\!36\)\( p^{18} T^{10} + \)\(12\!\cdots\!08\)\( p^{36} T^{12} - 1649449028392296 p^{54} T^{14} + p^{72} T^{16} \) | |
| 47 | \( ( 1 + 3716160 T + 1896060860372156 T^{2} + \)\(18\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!60\)\( p^{9} T^{5} + 1896060860372156 p^{18} T^{6} + 3716160 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 53 | \( 1 - 8766485591758824 T^{2} + \)\(43\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!80\)\( p^{18} T^{10} + \)\(43\!\cdots\!44\)\( p^{36} T^{12} - 8766485591758824 p^{54} T^{14} + p^{72} T^{16} \) | |
| 59 | \( 1 - 65444821078817512 T^{2} + \)\(19\!\cdots\!36\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{6} + \)\(34\!\cdots\!90\)\( T^{8} - \)\(32\!\cdots\!40\)\( p^{18} T^{10} + \)\(19\!\cdots\!36\)\( p^{36} T^{12} - 65444821078817512 p^{54} T^{14} + p^{72} T^{16} \) | |
| 61 | \( 1 - 33480681785208872 T^{2} + \)\(58\!\cdots\!36\)\( T^{4} - \)\(62\!\cdots\!40\)\( T^{6} + \)\(64\!\cdots\!50\)\( T^{8} - \)\(62\!\cdots\!40\)\( p^{18} T^{10} + \)\(58\!\cdots\!36\)\( p^{36} T^{12} - 33480681785208872 p^{54} T^{14} + p^{72} T^{16} \) | |
| 67 | \( 1 - 104404589351487656 T^{2} + \)\(63\!\cdots\!84\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(82\!\cdots\!50\)\( T^{8} - \)\(26\!\cdots\!20\)\( p^{18} T^{10} + \)\(63\!\cdots\!84\)\( p^{36} T^{12} - 104404589351487656 p^{54} T^{14} + p^{72} T^{16} \) | |
| 71 | \( ( 1 + 280117344 T + 145831167184353692 T^{2} + \)\(30\!\cdots\!96\)\( T^{3} + \)\(97\!\cdots\!34\)\( T^{4} + \)\(30\!\cdots\!96\)\( p^{9} T^{5} + 145831167184353692 p^{18} T^{6} + 280117344 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 261993560 T + 145417231313639804 T^{2} + \)\(33\!\cdots\!76\)\( T^{3} + \)\(99\!\cdots\!54\)\( T^{4} + \)\(33\!\cdots\!76\)\( p^{9} T^{5} + 145417231313639804 p^{18} T^{6} + 261993560 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 124471872 T + 432348011474041148 T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!42\)\( T^{4} - \)\(40\!\cdots\!20\)\( p^{9} T^{5} + 432348011474041148 p^{18} T^{6} - 124471872 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 83 | \( 1 - 620814363205633576 T^{2} + \)\(25\!\cdots\!08\)\( T^{4} - \)\(71\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!22\)\( T^{8} - \)\(71\!\cdots\!76\)\( p^{18} T^{10} + \)\(25\!\cdots\!08\)\( p^{36} T^{12} - 620814363205633576 p^{54} T^{14} + p^{72} T^{16} \) | |
| 89 | \( ( 1 - 372413928 T + 500734440582085948 T^{2} - \)\(19\!\cdots\!20\)\( p T^{3} + \)\(28\!\cdots\!22\)\( T^{4} - \)\(19\!\cdots\!20\)\( p^{10} T^{5} + 500734440582085948 p^{18} T^{6} - 372413928 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
| 97 | \( ( 1 + 4966392 T + 2046417225380869532 T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(19\!\cdots\!82\)\( T^{4} - \)\(22\!\cdots\!04\)\( p^{9} T^{5} + 2046417225380869532 p^{18} T^{6} + 4966392 p^{27} T^{7} + p^{36} T^{8} )^{2} \) | |
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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
−6.14306420006755495553527612793, −5.80905241946255655043608733167, −5.77868233801459276190698104920, −5.10591243378134567284336014425, −5.04194037412297533963599453400, −4.91581316253210606669518013231, −4.59317446781086374196315531315, −4.57586640106387947921755205031, −4.17705385911231739106549049906, −4.15236784411041784990037137539, −3.89777469448772688542432399075, −3.56112061057296311534502880973, −3.24646951447404293507292723616, −3.11700741377412760731543284127, −2.86802897061540713584200537744, −2.73089870420543675965833745513, −2.13344652311402317903482028210, −1.88962646569710632891945738573, −1.70606627750943089032332937619, −1.67521961233058835503693013938, −1.26218083819588726379720501182, −0.999353384134172904836315860687, −0.812801481558491617128610591567, −0.33558471761708097852046139717, −0.22758634972338549695041763341, 0.22758634972338549695041763341, 0.33558471761708097852046139717, 0.812801481558491617128610591567, 0.999353384134172904836315860687, 1.26218083819588726379720501182, 1.67521961233058835503693013938, 1.70606627750943089032332937619, 1.88962646569710632891945738573, 2.13344652311402317903482028210, 2.73089870420543675965833745513, 2.86802897061540713584200537744, 3.11700741377412760731543284127, 3.24646951447404293507292723616, 3.56112061057296311534502880973, 3.89777469448772688542432399075, 4.15236784411041784990037137539, 4.17705385911231739106549049906, 4.57586640106387947921755205031, 4.59317446781086374196315531315, 4.91581316253210606669518013231, 5.04194037412297533963599453400, 5.10591243378134567284336014425, 5.77868233801459276190698104920, 5.80905241946255655043608733167, 6.14306420006755495553527612793
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.