Dirichlet series
| L(s) = 1 | − 2-s + 2·7-s − 8-s − 6·11-s − 2·14-s − 3·16-s − 6·19-s + 6·22-s − 9·25-s + 16·29-s + 6·31-s + 6·32-s + 6·37-s + 6·38-s − 4·47-s + 4·49-s + 9·50-s + 4·53-s − 2·56-s − 16·58-s + 14·59-s + 12·61-s − 6·62-s − 64-s + 42·67-s − 18·73-s − 6·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.755·7-s − 0.353·8-s − 1.80·11-s − 0.534·14-s − 3/4·16-s − 1.37·19-s + 1.27·22-s − 9/5·25-s + 2.97·29-s + 1.07·31-s + 1.06·32-s + 0.986·37-s + 0.973·38-s − 0.583·47-s + 4/7·49-s + 1.27·50-s + 0.549·53-s − 0.267·56-s − 2.10·58-s + 1.82·59-s + 1.53·61-s − 0.762·62-s − 1/8·64-s + 5.13·67-s − 2.10·73-s − 0.697·74-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(268.794\) |
| Root analytic conductor: | \(1.41853\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9215683262\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9215683262\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + T + T^{2} + p T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 2 T - 16 T^{3} + 65 T^{4} - 16 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 + 9 T^{2} + 9 p T^{4} - 96 T^{5} + 66 T^{6} - 864 T^{7} - 394 T^{8} - 864 p T^{9} + 66 p^{2} T^{10} - 96 p^{3} T^{11} + 9 p^{5} T^{12} + 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T + 25 T^{2} - 58 T^{3} + 372 T^{4} - 58 p T^{5} + 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + 8 T + 34 T^{2} + 112 T^{3} + 339 T^{4} + 112 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \) | |
| 13 | \( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 49362 p^{2} T^{10} + 2241 p^{4} T^{12} - 66 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 40 T^{2} + 786 T^{4} - 1104 T^{5} + 10208 T^{6} - 39072 T^{7} + 129731 T^{8} - 39072 p T^{9} + 10208 p^{2} T^{10} - 1104 p^{3} T^{11} + 786 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 11868 p T^{9} - 34182 p^{2} T^{10} + 1968 p^{3} T^{11} + 1165 p^{4} T^{12} - 150 p^{5} T^{13} - 33 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 52 T^{2} + 54 p T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 140352 p T^{9} + 24080 p^{2} T^{10} - 2784 p^{3} T^{11} + 54 p^{5} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) | |
| 29 | \( ( 1 - 8 T + 71 T^{2} - 344 T^{3} + 1924 T^{4} - 344 p T^{5} + 71 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 31 | \( 1 - 6 T - 4 T^{2} + 336 T^{3} - 2729 T^{4} + 10764 T^{5} + 2216 T^{6} - 444234 T^{7} + 3877768 T^{8} - 444234 p T^{9} + 2216 p^{2} T^{10} + 10764 p^{3} T^{11} - 2729 p^{4} T^{12} + 336 p^{5} T^{13} - 4 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 146184 p T^{9} - 152586 p^{2} T^{10} + 9780 p^{3} T^{11} + 4753 p^{4} T^{12} - 18 p^{5} T^{13} - 69 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 607176 p^{2} T^{10} + 9948 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 1729746 p^{2} T^{10} + 23793 p^{4} T^{12} - 210 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 + 4 T - 144 T^{2} - 456 T^{3} + 12722 T^{4} + 27948 T^{5} - 805600 T^{6} - 556940 T^{7} + 41968563 T^{8} - 556940 p T^{9} - 805600 p^{2} T^{10} + 27948 p^{3} T^{11} + 12722 p^{4} T^{12} - 456 p^{5} T^{13} - 144 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 4 T - 135 T^{2} + 900 T^{3} + 9413 T^{4} - 70368 T^{5} - 312982 T^{6} + 1938608 T^{7} + 10598262 T^{8} + 1938608 p T^{9} - 312982 p^{2} T^{10} - 70368 p^{3} T^{11} + 9413 p^{4} T^{12} + 900 p^{5} T^{13} - 135 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 14 T - 13 T^{2} + 1110 T^{3} - 3463 T^{4} - 11848 T^{5} - 87914 T^{6} - 1416852 T^{7} + 32978194 T^{8} - 1416852 p T^{9} - 87914 p^{2} T^{10} - 11848 p^{3} T^{11} - 3463 p^{4} T^{12} + 1110 p^{5} T^{13} - 13 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 3576756 p T^{9} + 10032 p^{2} T^{10} - 46860 p^{3} T^{11} + 12426 p^{4} T^{12} - 1584 p^{5} T^{13} + 180 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 42 T + 1023 T^{2} - 18270 T^{3} + 261141 T^{4} - 3133152 T^{5} + 32837970 T^{6} - 307895844 T^{7} + 2631022010 T^{8} - 307895844 p T^{9} + 32837970 p^{2} T^{10} - 3133152 p^{3} T^{11} + 261141 p^{4} T^{12} - 18270 p^{5} T^{13} + 1023 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 4336608 p^{2} T^{10} + 41724 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 34967736 p T^{9} + 4384558 p^{2} T^{10} + 463140 p^{3} T^{11} + 50601 p^{4} T^{12} + 4302 p^{5} T^{13} + 347 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 6 T + 132 T^{2} + 720 T^{3} + 9999 T^{4} + 52644 T^{5} - 314880 T^{6} + 1496442 T^{7} - 48671848 T^{8} + 1496442 p T^{9} - 314880 p^{2} T^{10} + 52644 p^{3} T^{11} + 9999 p^{4} T^{12} + 720 p^{5} T^{13} + 132 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 802 p T^{5} + 229 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 264 T^{2} + 38610 T^{4} - 50400 T^{5} + 4052064 T^{6} - 8952768 T^{7} + 353995811 T^{8} - 8952768 p T^{9} + 4052064 p^{2} T^{10} - 50400 p^{3} T^{11} + 38610 p^{4} T^{12} + 264 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 28537554 p^{2} T^{10} + 165777 p^{4} T^{12} - 594 p^{6} T^{14} + p^{8} T^{16} \) | |
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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
−5.36372294756156158350282004443, −5.32286498477727899526715202230, −5.23820421538997933461192126831, −5.16577828931565750942510899931, −4.82965937439152877020118826160, −4.66021474433818028940933093985, −4.42936324390873685973407811378, −4.39753247320784581755008745118, −4.17959146924510661062528950724, −4.03846348595522515592845765906, −3.95327762730302146425938836115, −3.71752356856237051760454098476, −3.69760100500919173162550387944, −3.20106541827624145945970987185, −2.89180637606477860682756153213, −2.77013762647804974879285367635, −2.64606448723190940849797825502, −2.63619337065818592779357144419, −2.33264728578967249796656254934, −2.23483402693866479893864198210, −1.86023225272346241158289434204, −1.59156030031612830016084976914, −1.26262668655151211852510921938, −0.74415156774313237273644663285, −0.44221263707881791532500924964, 0.44221263707881791532500924964, 0.74415156774313237273644663285, 1.26262668655151211852510921938, 1.59156030031612830016084976914, 1.86023225272346241158289434204, 2.23483402693866479893864198210, 2.33264728578967249796656254934, 2.63619337065818592779357144419, 2.64606448723190940849797825502, 2.77013762647804974879285367635, 2.89180637606477860682756153213, 3.20106541827624145945970987185, 3.69760100500919173162550387944, 3.71752356856237051760454098476, 3.95327762730302146425938836115, 4.03846348595522515592845765906, 4.17959146924510661062528950724, 4.39753247320784581755008745118, 4.42936324390873685973407811378, 4.66021474433818028940933093985, 4.82965937439152877020118826160, 5.16577828931565750942510899931, 5.23820421538997933461192126831, 5.32286498477727899526715202230, 5.36372294756156158350282004443
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.