Dirichlet series
| L(s) = 1 | + 8·2-s + 32·4-s + 80·8-s + 16·11-s − 8·13-s + 120·16-s + 8·17-s + 128·22-s + 40·23-s − 64·26-s − 96·31-s + 32·32-s + 64·34-s − 80·37-s + 32·41-s + 80·43-s + 512·44-s + 320·46-s + 112·47-s − 256·52-s − 112·53-s + 256·61-s − 768·62-s − 384·64-s + 128·67-s + 256·68-s − 384·71-s + ⋯ |
| L(s) = 1 | + 4·2-s + 8·4-s + 10·8-s + 1.45·11-s − 0.615·13-s + 15/2·16-s + 8/17·17-s + 5.81·22-s + 1.73·23-s − 2.46·26-s − 3.09·31-s + 32-s + 1.88·34-s − 2.16·37-s + 0.780·41-s + 1.86·43-s + 11.6·44-s + 6.95·46-s + 2.38·47-s − 4.92·52-s − 2.11·53-s + 4.19·61-s − 12.3·62-s − 6·64-s + 1.91·67-s + 3.76·68-s − 5.40·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48946\times 10^{11}\) |
| Root analytic conductor: | \(5.34887\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(4.104944100\) |
| \(L(\frac12)\) | \(\approx\) | \(4.104944100\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| 5 | \( 1 \) | |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) | |
| good | 11 | \( ( 1 - 8 T + 30 p T^{2} - 1216 T^{3} + 47051 T^{4} - 1216 p^{2} T^{5} + 30 p^{5} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 + 8 T + 32 T^{2} - 664 T^{3} - 49652 T^{4} - 98344 T^{5} + 1022560 T^{6} + 41872056 T^{7} + 1725329958 T^{8} + 41872056 p^{2} T^{9} + 1022560 p^{4} T^{10} - 98344 p^{6} T^{11} - 49652 p^{8} T^{12} - 664 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 - 8 T + 32 T^{2} - 6376 T^{3} + 21068 T^{4} + 2037224 T^{5} + 3354720 T^{6} + 258309384 T^{7} - 16247435162 T^{8} + 258309384 p^{2} T^{9} + 3354720 p^{4} T^{10} + 2037224 p^{6} T^{11} + 21068 p^{8} T^{12} - 6376 p^{10} T^{13} + 32 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 - 1656 T^{2} + 1278484 T^{4} - 647922024 T^{6} + 256480581990 T^{8} - 647922024 p^{4} T^{10} + 1278484 p^{8} T^{12} - 1656 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 - 40 T + 800 T^{2} - 27280 T^{3} + 416398 T^{4} + 5986520 T^{5} - 200480000 T^{6} + 9108994920 T^{7} - 354673135437 T^{8} + 9108994920 p^{2} T^{9} - 200480000 p^{4} T^{10} + 5986520 p^{6} T^{11} + 416398 p^{8} T^{12} - 27280 p^{10} T^{13} + 800 p^{12} T^{14} - 40 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 4204 T^{2} + 9385738 T^{4} - 13413584176 T^{6} + 13417559903347 T^{8} - 13413584176 p^{4} T^{10} + 9385738 p^{8} T^{12} - 4204 p^{12} T^{14} + p^{16} T^{16} \) | |
| 31 | \( ( 1 + 48 T + 4380 T^{2} + 140112 T^{3} + 6571718 T^{4} + 140112 p^{2} T^{5} + 4380 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 37 | \( 1 + 80 T + 3200 T^{2} + 78944 T^{3} + 424606 T^{4} - 92138992 T^{5} - 5613780992 T^{6} - 235665534864 T^{7} - 8520280609245 T^{8} - 235665534864 p^{2} T^{9} - 5613780992 p^{4} T^{10} - 92138992 p^{6} T^{11} + 424606 p^{8} T^{12} + 78944 p^{10} T^{13} + 3200 p^{12} T^{14} + 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( ( 1 - 16 T + 3588 T^{2} - 87344 T^{3} + 7589702 T^{4} - 87344 p^{2} T^{5} + 3588 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 43 | \( 1 - 80 T + 3200 T^{2} - 109280 T^{3} + 7332142 T^{4} - 484105040 T^{5} + 21236608000 T^{6} - 733663929840 T^{7} + 24238639221939 T^{8} - 733663929840 p^{2} T^{9} + 21236608000 p^{4} T^{10} - 484105040 p^{6} T^{11} + 7332142 p^{8} T^{12} - 109280 p^{10} T^{13} + 3200 p^{12} T^{14} - 80 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 - 112 T + 6272 T^{2} - 443968 T^{3} + 40120780 T^{4} - 2265671968 T^{5} + 100671520768 T^{6} - 5937892291632 T^{7} + 341467810934118 T^{8} - 5937892291632 p^{2} T^{9} + 100671520768 p^{4} T^{10} - 2265671968 p^{6} T^{11} + 40120780 p^{8} T^{12} - 443968 p^{10} T^{13} + 6272 p^{12} T^{14} - 112 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( 1 + 112 T + 6272 T^{2} + 293104 T^{3} + 27071044 T^{4} + 2319918160 T^{5} + 132996223360 T^{6} + 6340658486736 T^{7} + 302065305633606 T^{8} + 6340658486736 p^{2} T^{9} + 132996223360 p^{4} T^{10} + 2319918160 p^{6} T^{11} + 27071044 p^{8} T^{12} + 293104 p^{10} T^{13} + 6272 p^{12} T^{14} + 112 p^{14} T^{15} + p^{16} T^{16} \) | |
| 59 | \( 1 - 14520 T^{2} + 112073620 T^{4} - 598743129384 T^{6} + 2383933474013670 T^{8} - 598743129384 p^{4} T^{10} + 112073620 p^{8} T^{12} - 14520 p^{12} T^{14} + p^{16} T^{16} \) | |
| 61 | \( ( 1 - 128 T + 16548 T^{2} - 1273216 T^{3} + 92120294 T^{4} - 1273216 p^{2} T^{5} + 16548 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 67 | \( 1 - 128 T + 8192 T^{2} - 1209152 T^{3} + 93124990 T^{4} - 3011796032 T^{5} + 353654253568 T^{6} - 20812676382528 T^{7} + 195235083944163 T^{8} - 20812676382528 p^{2} T^{9} + 353654253568 p^{4} T^{10} - 3011796032 p^{6} T^{11} + 93124990 p^{8} T^{12} - 1209152 p^{10} T^{13} + 8192 p^{12} T^{14} - 128 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( ( 1 + 192 T + 24882 T^{2} + 2523216 T^{3} + 199316435 T^{4} + 2523216 p^{2} T^{5} + 24882 p^{4} T^{6} + 192 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 73 | \( 1 - 168 T + 14112 T^{2} - 1165416 T^{3} + 103108684 T^{4} - 8788754520 T^{5} + 700538237280 T^{6} - 64210806364824 T^{7} + 5584886034801126 T^{8} - 64210806364824 p^{2} T^{9} + 700538237280 p^{4} T^{10} - 8788754520 p^{6} T^{11} + 103108684 p^{8} T^{12} - 1165416 p^{10} T^{13} + 14112 p^{12} T^{14} - 168 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( 1 - 25916 T^{2} + 346430202 T^{4} - 3284221296400 T^{6} + 23601534235814051 T^{8} - 3284221296400 p^{4} T^{10} + 346430202 p^{8} T^{12} - 25916 p^{12} T^{14} + p^{16} T^{16} \) | |
| 83 | \( 1 - 64 T + 2048 T^{2} - 334144 T^{3} + 107854276 T^{4} - 5878080448 T^{5} + 211137697792 T^{6} - 39218853943488 T^{7} + 7265503548957126 T^{8} - 39218853943488 p^{2} T^{9} + 211137697792 p^{4} T^{10} - 5878080448 p^{6} T^{11} + 107854276 p^{8} T^{12} - 334144 p^{10} T^{13} + 2048 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 - 9944 T^{2} + 91408788 T^{4} - 912387941896 T^{6} + 9802062914880422 T^{8} - 912387941896 p^{4} T^{10} + 91408788 p^{8} T^{12} - 9944 p^{12} T^{14} + p^{16} T^{16} \) | |
| 97 | \( 1 + 112 T + 6272 T^{2} - 617680 T^{3} - 88063804 T^{4} + 3539173904 T^{5} + 1139487947136 T^{6} + 63717560038608 T^{7} + 5344624131812230 T^{8} + 63717560038608 p^{2} T^{9} + 1139487947136 p^{4} T^{10} + 3539173904 p^{6} T^{11} - 88063804 p^{8} T^{12} - 617680 p^{10} T^{13} + 6272 p^{12} T^{14} + 112 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−3.97789242955745146837982572914, −3.93021376278150872026089086697, −3.92484247224260023992416402166, −3.60874426361372676062235108951, −3.52429046549492475064068221837, −3.48569076266343627383830098052, −3.47090924733130791578828754222, −3.27068301014435986056341512412, −2.90777990986178746544405967212, −2.85199305244581226980735218576, −2.83615274239276271120834727282, −2.61109063775357877117249040851, −2.54064953851808715784935549802, −2.42413032995908216154318046241, −2.07486301969018426673209241494, −2.04435872034317663488603505427, −1.82157428717825222004413323435, −1.81878312268026348247954824861, −1.41244071411608078749574631347, −1.26281723684873744086251884569, −1.10703821071159329308184685640, −0.902567247685052145380207870590, −0.809919456820545597205641282224, −0.27067370279907990673749132050, −0.06049289509876530047087115225, 0.06049289509876530047087115225, 0.27067370279907990673749132050, 0.809919456820545597205641282224, 0.902567247685052145380207870590, 1.10703821071159329308184685640, 1.26281723684873744086251884569, 1.41244071411608078749574631347, 1.81878312268026348247954824861, 1.82157428717825222004413323435, 2.04435872034317663488603505427, 2.07486301969018426673209241494, 2.42413032995908216154318046241, 2.54064953851808715784935549802, 2.61109063775357877117249040851, 2.83615274239276271120834727282, 2.85199305244581226980735218576, 2.90777990986178746544405967212, 3.27068301014435986056341512412, 3.47090924733130791578828754222, 3.48569076266343627383830098052, 3.52429046549492475064068221837, 3.60874426361372676062235108951, 3.92484247224260023992416402166, 3.93021376278150872026089086697, 3.97789242955745146837982572914
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.