Dirichlet series
| L(s) = 1 | + 4-s − 10·11-s − 20·13-s − 10·17-s − 8·19-s + 10·23-s − 5·25-s + 22·29-s + 24·31-s − 20·37-s − 22·41-s − 10·44-s − 10·47-s + 32·49-s − 20·52-s + 30·53-s + 20·59-s + 10·67-s − 10·68-s − 20·71-s − 20·73-s − 8·76-s + 16·79-s − 70·83-s + 34·89-s + 10·92-s + 60·97-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.01·11-s − 5.54·13-s − 2.42·17-s − 1.83·19-s + 2.08·23-s − 25-s + 4.08·29-s + 4.31·31-s − 3.28·37-s − 3.43·41-s − 1.50·44-s − 1.45·47-s + 32/7·49-s − 2.77·52-s + 4.12·53-s + 2.60·59-s + 1.22·67-s − 1.21·68-s − 2.37·71-s − 2.34·73-s − 0.917·76-s + 1.80·79-s − 7.68·83-s + 3.60·89-s + 1.04·92-s + 6.09·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(27791.8\) |
| Root analytic conductor: | \(1.89559\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9718746697\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9718746697\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 + p T^{2} - 2 p T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{8} \) | |
| good | 7 | \( 1 - 32 T^{2} + 500 T^{4} - 736 p T^{6} + 40294 T^{8} - 736 p^{3} T^{10} + 500 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 10 T + 28 T^{2} - 60 T^{3} - 617 T^{4} - 1790 T^{5} - 794 T^{6} + 18400 T^{7} + 94725 T^{8} + 18400 p T^{9} - 794 p^{2} T^{10} - 1790 p^{3} T^{11} - 617 p^{4} T^{12} - 60 p^{5} T^{13} + 28 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 20 T + 202 T^{2} + 1420 T^{3} + 8095 T^{4} + 40660 T^{5} + 14284 p T^{6} + 772160 T^{7} + 2918789 T^{8} + 772160 p T^{9} + 14284 p^{3} T^{10} + 40660 p^{3} T^{11} + 8095 p^{4} T^{12} + 1420 p^{5} T^{13} + 202 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 10 T + 78 T^{2} + 590 T^{3} + 3615 T^{4} + 20330 T^{5} + 105768 T^{6} + 488740 T^{7} + 2089449 T^{8} + 488740 p T^{9} + 105768 p^{2} T^{10} + 20330 p^{3} T^{11} + 3615 p^{4} T^{12} + 590 p^{5} T^{13} + 78 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 8 T - 10 T^{2} - 270 T^{3} - 565 T^{4} + 5224 T^{5} + 28422 T^{6} - 2940 p T^{7} - 41505 p T^{8} - 2940 p^{2} T^{9} + 28422 p^{2} T^{10} + 5224 p^{3} T^{11} - 565 p^{4} T^{12} - 270 p^{5} T^{13} - 10 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 10 T + 116 T^{2} - 950 T^{3} + 6927 T^{4} - 45910 T^{5} + 11616 p T^{6} - 1460600 T^{7} + 7269605 T^{8} - 1460600 p T^{9} + 11616 p^{3} T^{10} - 45910 p^{3} T^{11} + 6927 p^{4} T^{12} - 950 p^{5} T^{13} + 116 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 22 T + 10 p T^{2} - 2930 T^{3} + 24395 T^{4} - 175526 T^{5} + 1131792 T^{6} - 6698140 T^{7} + 37055745 T^{8} - 6698140 p T^{9} + 1131792 p^{2} T^{10} - 175526 p^{3} T^{11} + 24395 p^{4} T^{12} - 2930 p^{5} T^{13} + 10 p^{7} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 24 T + 230 T^{2} - 990 T^{3} - 685 T^{4} + 32568 T^{5} - 115202 T^{6} - 726180 T^{7} + 8308045 T^{8} - 726180 p T^{9} - 115202 p^{2} T^{10} + 32568 p^{3} T^{11} - 685 p^{4} T^{12} - 990 p^{5} T^{13} + 230 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 20 T + 275 T^{2} + 3140 T^{3} + 29831 T^{4} + 251620 T^{5} + 1934025 T^{6} + 13441540 T^{7} + 85392936 T^{8} + 13441540 p T^{9} + 1934025 p^{2} T^{10} + 251620 p^{3} T^{11} + 29831 p^{4} T^{12} + 3140 p^{5} T^{13} + 275 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 22 T + 146 T^{2} - 142 T^{3} - 6829 T^{4} - 52634 T^{5} - 236776 T^{6} + 967444 T^{7} + 17720057 T^{8} + 967444 p T^{9} - 236776 p^{2} T^{10} - 52634 p^{3} T^{11} - 6829 p^{4} T^{12} - 142 p^{5} T^{13} + 146 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 120 T^{2} + 10856 T^{4} - 676920 T^{6} + 33149566 T^{8} - 676920 p^{2} T^{10} + 10856 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 + 10 T + 164 T^{2} + 1060 T^{3} + 6767 T^{4} + 10550 T^{5} - 301398 T^{6} - 3376520 T^{7} - 34256075 T^{8} - 3376520 p T^{9} - 301398 p^{2} T^{10} + 10550 p^{3} T^{11} + 6767 p^{4} T^{12} + 1060 p^{5} T^{13} + 164 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 30 T + 425 T^{2} - 3090 T^{3} + 5031 T^{4} + 120570 T^{5} - 1120625 T^{6} + 3254310 T^{7} + 1334576 T^{8} + 3254310 p T^{9} - 1120625 p^{2} T^{10} + 120570 p^{3} T^{11} + 5031 p^{4} T^{12} - 3090 p^{5} T^{13} + 425 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 20 T + 242 T^{2} - 2760 T^{3} + 31483 T^{4} - 335220 T^{5} + 3074684 T^{6} - 25371200 T^{7} + 201486805 T^{8} - 25371200 p T^{9} + 3074684 p^{2} T^{10} - 335220 p^{3} T^{11} + 31483 p^{4} T^{12} - 2760 p^{5} T^{13} + 242 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 118 T^{2} + 10203 T^{4} + 835676 T^{6} + 64821605 T^{8} + 835676 p^{2} T^{10} + 10203 p^{4} T^{12} + 118 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 - 10 T + 88 T^{2} - 1660 T^{3} + 11615 T^{4} - 86790 T^{5} + 911158 T^{6} - 5793400 T^{7} + 42525189 T^{8} - 5793400 p T^{9} + 911158 p^{2} T^{10} - 86790 p^{3} T^{11} + 11615 p^{4} T^{12} - 1660 p^{5} T^{13} + 88 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 20 T + 118 T^{2} - 490 T^{3} - 14397 T^{4} - 149580 T^{5} - 402414 T^{6} + 9696700 T^{7} + 137916525 T^{8} + 9696700 p T^{9} - 402414 p^{2} T^{10} - 149580 p^{3} T^{11} - 14397 p^{4} T^{12} - 490 p^{5} T^{13} + 118 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 20 T + 282 T^{2} + 3780 T^{3} + 46315 T^{4} + 505780 T^{5} + 5254892 T^{6} + 48564720 T^{7} + 416447549 T^{8} + 48564720 p T^{9} + 5254892 p^{2} T^{10} + 505780 p^{3} T^{11} + 46315 p^{4} T^{12} + 3780 p^{5} T^{13} + 282 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 16 T + 274 T^{2} - 44 p T^{3} + 48131 T^{4} - 506768 T^{5} + 5479676 T^{6} - 51684448 T^{7} + 496487797 T^{8} - 51684448 p T^{9} + 5479676 p^{2} T^{10} - 506768 p^{3} T^{11} + 48131 p^{4} T^{12} - 44 p^{6} T^{13} + 274 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 70 T + 2432 T^{2} + 56730 T^{3} + 1010935 T^{4} + 14768130 T^{5} + 184247072 T^{6} + 2010015520 T^{7} + 19419043189 T^{8} + 2010015520 p T^{9} + 184247072 p^{2} T^{10} + 14768130 p^{3} T^{11} + 1010935 p^{4} T^{12} + 56730 p^{5} T^{13} + 2432 p^{6} T^{14} + 70 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 34 T + 409 T^{2} - 1014 T^{3} - 27449 T^{4} + 344318 T^{5} - 919769 T^{6} - 22713862 T^{7} + 350578752 T^{8} - 22713862 p T^{9} - 919769 p^{2} T^{10} + 344318 p^{3} T^{11} - 27449 p^{4} T^{12} - 1014 p^{5} T^{13} + 409 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 60 T + 1990 T^{2} - 47940 T^{3} + 920391 T^{4} - 14754060 T^{5} + 202723580 T^{6} - 2424996600 T^{7} + 25463108381 T^{8} - 2424996600 p T^{9} + 202723580 p^{2} T^{10} - 14754060 p^{3} T^{11} + 920391 p^{4} T^{12} - 47940 p^{5} T^{13} + 1990 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.84791362889025501256095932220, −4.80033483738633088175355139718, −4.75160734853668371098797606249, −4.51555352893640220070386060470, −4.49211498639220447791755919303, −4.36625168588765561632808297057, −4.05837340760767727043769123992, −4.03952480245351232186578415172, −3.72667450844573541060004271184, −3.58567103028615275454632028059, −3.19036534963291407526264812365, −2.98811626195705358297025485426, −2.94903789574831393194510162154, −2.76674852979827351002450562637, −2.68556387787147667823323273572, −2.53800543652310987743690125520, −2.42441790941694410586932354657, −2.26438542943697736968498208351, −2.17116850901018209973499477455, −2.08851893699832513749123309165, −1.77404131112848282278957877911, −1.34205802167370427709055950119, −0.71487826075278723959125970772, −0.53978352746536310044827721819, −0.31483236115905909042509069562, 0.31483236115905909042509069562, 0.53978352746536310044827721819, 0.71487826075278723959125970772, 1.34205802167370427709055950119, 1.77404131112848282278957877911, 2.08851893699832513749123309165, 2.17116850901018209973499477455, 2.26438542943697736968498208351, 2.42441790941694410586932354657, 2.53800543652310987743690125520, 2.68556387787147667823323273572, 2.76674852979827351002450562637, 2.94903789574831393194510162154, 2.98811626195705358297025485426, 3.19036534963291407526264812365, 3.58567103028615275454632028059, 3.72667450844573541060004271184, 4.03952480245351232186578415172, 4.05837340760767727043769123992, 4.36625168588765561632808297057, 4.49211498639220447791755919303, 4.51555352893640220070386060470, 4.75160734853668371098797606249, 4.80033483738633088175355139718, 4.84791362889025501256095932220
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.