Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·5-s + 4·6-s − 2·7-s + 9-s − 8·10-s − 2·12-s + 4·13-s + 4·14-s − 8·15-s − 8·17-s − 2·18-s − 12·19-s + 4·20-s + 4·21-s − 4·23-s + 15·25-s − 8·26-s − 2·28-s − 4·29-s + 16·30-s − 14·31-s + 2·32-s + 16·34-s − 8·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.78·5-s + 1.63·6-s − 0.755·7-s + 1/3·9-s − 2.52·10-s − 0.577·12-s + 1.10·13-s + 1.06·14-s − 2.06·15-s − 1.94·17-s − 0.471·18-s − 2.75·19-s + 0.894·20-s + 0.872·21-s − 0.834·23-s + 3·25-s − 1.56·26-s − 0.377·28-s − 0.742·29-s + 2.92·30-s − 2.51·31-s + 0.353·32-s + 2.74·34-s − 1.35·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(34304.6\) |
| Root analytic conductor: | \(1.92070\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4601380202\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4601380202\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 29 T^{2} + 41 p T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 4 T + T^{2} + 6 T^{3} + T^{4} - 2 p T^{5} + 47 p T^{6} - 16 p^{2} T^{7} - 12 p^{2} T^{8} - 16 p^{3} T^{9} + 47 p^{3} T^{10} - 2 p^{4} T^{11} + p^{4} T^{12} + 6 p^{5} T^{13} + p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 - 4 T - 8 T^{2} + 6 T^{3} + 347 T^{4} - 376 T^{5} - 3724 T^{6} + 2372 T^{7} + 42877 T^{8} + 2372 p T^{9} - 3724 p^{2} T^{10} - 376 p^{3} T^{11} + 347 p^{4} T^{12} + 6 p^{5} T^{13} - 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 8 T + 70 T^{2} + 464 T^{3} + 2871 T^{4} + 14952 T^{5} + 4548 p T^{6} + 346600 T^{7} + 1499829 T^{8} + 346600 p T^{9} + 4548 p^{3} T^{10} + 14952 p^{3} T^{11} + 2871 p^{4} T^{12} + 464 p^{5} T^{13} + 70 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 12 T + 31 T^{2} - 96 T^{3} - 5 p T^{4} + 2512 T^{5} + 4105 T^{6} + 260 p T^{7} + 101276 T^{8} + 260 p^{2} T^{9} + 4105 p^{2} T^{10} + 2512 p^{3} T^{11} - 5 p^{5} T^{12} - 96 p^{5} T^{13} + 31 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + 2 T + 83 T^{2} + 6 p T^{3} + 2765 T^{4} + 6 p^{2} T^{5} + 83 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 4 T - 12 T^{2} - 334 T^{3} - 677 T^{4} + 4024 T^{5} + 31110 T^{6} + 4800 T^{7} - 490599 T^{8} + 4800 p T^{9} + 31110 p^{2} T^{10} + 4024 p^{3} T^{11} - 677 p^{4} T^{12} - 334 p^{5} T^{13} - 12 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 14 T + 55 T^{2} - 70 T^{3} - 545 T^{4} + 3802 T^{5} + 913 T^{6} - 358610 T^{7} - 2899360 T^{8} - 358610 p T^{9} + 913 p^{2} T^{10} + 3802 p^{3} T^{11} - 545 p^{4} T^{12} - 70 p^{5} T^{13} + 55 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 10 T - 20 T^{2} + 530 T^{3} - 419 T^{4} - 530 T^{5} - 101000 T^{6} - 176960 T^{7} + 7183361 T^{8} - 176960 p T^{9} - 101000 p^{2} T^{10} - 530 p^{3} T^{11} - 419 p^{4} T^{12} + 530 p^{5} T^{13} - 20 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 12 T - 38 T^{2} - 1388 T^{3} - 5157 T^{4} + 59852 T^{5} + 529180 T^{6} - 958480 T^{7} - 26289619 T^{8} - 958480 p T^{9} + 529180 p^{2} T^{10} + 59852 p^{3} T^{11} - 5157 p^{4} T^{12} - 1388 p^{5} T^{13} - 38 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 10 T + 100 T^{2} - 630 T^{3} + 5018 T^{4} - 630 p T^{5} + 100 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 28 T + 328 T^{2} - 1882 T^{3} + 5667 T^{4} - 21792 T^{5} - 27606 T^{6} + 4499416 T^{7} - 50513343 T^{8} + 4499416 p T^{9} - 27606 p^{2} T^{10} - 21792 p^{3} T^{11} + 5667 p^{4} T^{12} - 1882 p^{5} T^{13} + 328 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T - 108 T^{2} + 334 T^{3} + 4431 T^{4} - 20942 T^{5} + 68130 T^{6} + 576352 T^{7} - 13977591 T^{8} + 576352 p T^{9} + 68130 p^{2} T^{10} - 20942 p^{3} T^{11} + 4431 p^{4} T^{12} + 334 p^{5} T^{13} - 108 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 128 T^{2} + 10 p T^{3} + 4403 T^{4} - 64500 T^{5} + 495534 T^{6} + 2274000 T^{7} - 52578895 T^{8} + 2274000 p T^{9} + 495534 p^{2} T^{10} - 64500 p^{3} T^{11} + 4403 p^{4} T^{12} + 10 p^{6} T^{13} - 128 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 + 34 T + 440 T^{2} + 2590 T^{3} + 11615 T^{4} + 181262 T^{5} + 2141538 T^{6} + 10457720 T^{7} + 32037945 T^{8} + 10457720 p T^{9} + 2141538 p^{2} T^{10} + 181262 p^{3} T^{11} + 11615 p^{4} T^{12} + 2590 p^{5} T^{13} + 440 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 12 T + 210 T^{2} + 1848 T^{3} + 19566 T^{4} + 1848 p T^{5} + 210 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 98 T^{2} - 1180 T^{3} + 2723 T^{4} + 115440 T^{5} + 800604 T^{6} - 4163760 T^{7} - 89441035 T^{8} - 4163760 p T^{9} + 800604 p^{2} T^{10} + 115440 p^{3} T^{11} + 2723 p^{4} T^{12} - 1180 p^{5} T^{13} - 98 p^{6} T^{14} + p^{8} T^{16} \) | |
| 73 | \( 1 - 60 T^{2} - 130 T^{3} + 4971 T^{4} + 47100 T^{5} - 144550 T^{6} - 297320 T^{7} + 13172041 T^{8} - 297320 p T^{9} - 144550 p^{2} T^{10} + 47100 p^{3} T^{11} + 4971 p^{4} T^{12} - 130 p^{5} T^{13} - 60 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( 1 - 22 T + 226 T^{2} - 1904 T^{3} + 19415 T^{4} - 141262 T^{5} + 470990 T^{6} - 1200080 T^{7} + 20190961 T^{8} - 1200080 p T^{9} + 470990 p^{2} T^{10} - 141262 p^{3} T^{11} + 19415 p^{4} T^{12} - 1904 p^{5} T^{13} + 226 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 30 T + 580 T^{2} + 8810 T^{3} + 117311 T^{4} + 1406610 T^{5} + 15300330 T^{6} + 153332760 T^{7} + 1430690321 T^{8} + 153332760 p T^{9} + 15300330 p^{2} T^{10} + 1406610 p^{3} T^{11} + 117311 p^{4} T^{12} + 8810 p^{5} T^{13} + 580 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 2 T + 47 T^{2} + 4 T^{3} + 13249 T^{4} + 4 p T^{5} + 47 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 10 T - 140 T^{2} - 810 T^{3} + 25071 T^{4} + 100030 T^{5} - 2624870 T^{6} + 791440 T^{7} + 357916921 T^{8} + 791440 p T^{9} - 2624870 p^{2} T^{10} + 100030 p^{3} T^{11} + 25071 p^{4} T^{12} - 810 p^{5} T^{13} - 140 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.89722075626651762947835647334, −4.68229232468318254222525898269, −4.59404907704110533908284828906, −4.47284360437218767303202342343, −4.29364122228638603756250219864, −4.26020494526862486560584087304, −4.14852514782513353695694717773, −3.84599303749077107244115526535, −3.72547445798622562159639928213, −3.67400440816457129355981211306, −3.26605840146228804163506497286, −3.15856692873487934381894599146, −3.01832185698129161496242803680, −2.75445705703679469998216964324, −2.62722582157417563419684918311, −2.53257925801006396668733239591, −2.17005632878693401545062499304, −2.01545090936526859601004338116, −1.92016952337529232773148451986, −1.71554922661605547441226596391, −1.69708343738000190971245656694, −1.23227677090498508388998701656, −0.76812886309882771550959494721, −0.57834052017117307490216642339, −0.28761356283273995710718912295, 0.28761356283273995710718912295, 0.57834052017117307490216642339, 0.76812886309882771550959494721, 1.23227677090498508388998701656, 1.69708343738000190971245656694, 1.71554922661605547441226596391, 1.92016952337529232773148451986, 2.01545090936526859601004338116, 2.17005632878693401545062499304, 2.53257925801006396668733239591, 2.62722582157417563419684918311, 2.75445705703679469998216964324, 3.01832185698129161496242803680, 3.15856692873487934381894599146, 3.26605840146228804163506497286, 3.67400440816457129355981211306, 3.72547445798622562159639928213, 3.84599303749077107244115526535, 4.14852514782513353695694717773, 4.26020494526862486560584087304, 4.29364122228638603756250219864, 4.47284360437218767303202342343, 4.59404907704110533908284828906, 4.68229232468318254222525898269, 4.89722075626651762947835647334
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.