Dirichlet series
| L(s) = 1 | + 2·4-s − 6·11-s + 12·13-s + 16-s − 2·17-s − 12·19-s − 8·23-s + 8·25-s − 2·29-s + 18·37-s − 12·41-s − 8·43-s − 12·44-s + 2·49-s + 24·52-s + 12·53-s − 18·59-s − 8·61-s − 2·64-s + 18·67-s − 4·68-s − 6·71-s − 24·76-s − 4·79-s + 18·89-s − 16·92-s − 54·97-s + ⋯ |
| L(s) = 1 | + 4-s − 1.80·11-s + 3.32·13-s + 1/4·16-s − 0.485·17-s − 2.75·19-s − 1.66·23-s + 8/5·25-s − 0.371·29-s + 2.95·37-s − 1.87·41-s − 1.21·43-s − 1.80·44-s + 2/7·49-s + 3.32·52-s + 1.64·53-s − 2.34·59-s − 1.02·61-s − 1/4·64-s + 2.19·67-s − 0.485·68-s − 0.712·71-s − 2.75·76-s − 0.450·79-s + 1.90·89-s − 1.66·92-s − 5.48·97-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.56501\times 10^{8}\) |
| Root analytic conductor: | \(3.61655\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.921792251\) |
| \(L(\frac12)\) | \(\approx\) | \(1.921792251\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 - T^{2} + T^{4} )^{2} \) | |
| 13 | \( 1 - 12 T + 66 T^{2} - 240 T^{3} + 815 T^{4} - 240 p T^{5} + 66 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( 1 - 8 T^{2} + 16 T^{4} + 88 T^{6} - 866 T^{8} + 88 p^{2} T^{10} + 16 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 6 T + 47 T^{2} + 210 T^{3} + 981 T^{4} + 3024 T^{5} + 11410 T^{6} + 29100 T^{7} + 109418 T^{8} + 29100 p T^{9} + 11410 p^{2} T^{10} + 3024 p^{3} T^{11} + 981 p^{4} T^{12} + 210 p^{5} T^{13} + 47 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 2 T - 2 p T^{2} - 84 T^{3} + 377 T^{4} + 772 T^{5} - 8822 T^{6} + 990 T^{7} + 252316 T^{8} + 990 p T^{9} - 8822 p^{2} T^{10} + 772 p^{3} T^{11} + 377 p^{4} T^{12} - 84 p^{5} T^{13} - 2 p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 12 T + 79 T^{2} + 372 T^{3} + 1173 T^{4} + 3456 T^{5} + 12518 T^{6} + 54120 T^{7} + 273206 T^{8} + 54120 p T^{9} + 12518 p^{2} T^{10} + 3456 p^{3} T^{11} + 1173 p^{4} T^{12} + 372 p^{5} T^{13} + 79 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 8 T - 13 T^{2} - 24 T^{3} + 1313 T^{4} - 896 T^{5} - 33326 T^{6} - 52320 T^{7} + 13030 T^{8} - 52320 p T^{9} - 33326 p^{2} T^{10} - 896 p^{3} T^{11} + 1313 p^{4} T^{12} - 24 p^{5} T^{13} - 13 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T - 81 T^{2} - 210 T^{3} + 3533 T^{4} + 8412 T^{5} - 113470 T^{6} - 115672 T^{7} + 3334050 T^{8} - 115672 p T^{9} - 113470 p^{2} T^{10} + 8412 p^{3} T^{11} + 3533 p^{4} T^{12} - 210 p^{5} T^{13} - 81 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 134 T^{2} + 9957 T^{4} - 498742 T^{6} + 17969120 T^{8} - 498742 p^{2} T^{10} + 9957 p^{4} T^{12} - 134 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 18 T + 262 T^{2} - 2772 T^{3} + 25822 T^{4} - 208494 T^{5} + 1563496 T^{6} - 10593630 T^{7} + 67601779 T^{8} - 10593630 p T^{9} + 1563496 p^{2} T^{10} - 208494 p^{3} T^{11} + 25822 p^{4} T^{12} - 2772 p^{5} T^{13} + 262 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 12 T + 183 T^{2} + 1620 T^{3} + 16501 T^{4} + 141000 T^{5} + 1092870 T^{6} + 8035728 T^{7} + 50331774 T^{8} + 8035728 p T^{9} + 1092870 p^{2} T^{10} + 141000 p^{3} T^{11} + 16501 p^{4} T^{12} + 1620 p^{5} T^{13} + 183 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 8 T - 83 T^{2} - 740 T^{3} + 113 p T^{4} + 36644 T^{5} - 203824 T^{6} - 596168 T^{7} + 9894598 T^{8} - 596168 p T^{9} - 203824 p^{2} T^{10} + 36644 p^{3} T^{11} + 113 p^{5} T^{12} - 740 p^{5} T^{13} - 83 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 2 T^{2} + 6297 T^{4} + 30106 T^{6} + 18959060 T^{8} + 30106 p^{2} T^{10} + 6297 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( ( 1 - 6 T + 2 p T^{2} - 528 T^{3} + 8079 T^{4} - 528 p T^{5} + 2 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 + 18 T + 263 T^{2} + 2790 T^{3} + 23073 T^{4} + 199872 T^{5} + 1712590 T^{6} + 14581596 T^{7} + 125880458 T^{8} + 14581596 p T^{9} + 1712590 p^{2} T^{10} + 199872 p^{3} T^{11} + 23073 p^{4} T^{12} + 2790 p^{5} T^{13} + 263 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 8 T - 62 T^{2} + 304 T^{3} + 6641 T^{4} - 36880 T^{5} + 81026 T^{6} + 2415160 T^{7} - 10221452 T^{8} + 2415160 p T^{9} + 81026 p^{2} T^{10} - 36880 p^{3} T^{11} + 6641 p^{4} T^{12} + 304 p^{5} T^{13} - 62 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 18 T + 267 T^{2} - 2862 T^{3} + 23857 T^{4} - 114264 T^{5} - 1314 T^{6} + 7237260 T^{7} - 85368366 T^{8} + 7237260 p T^{9} - 1314 p^{2} T^{10} - 114264 p^{3} T^{11} + 23857 p^{4} T^{12} - 2862 p^{5} T^{13} + 267 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 6 T + 257 T^{2} + 1470 T^{3} + 38071 T^{4} + 238032 T^{5} + 4008512 T^{6} + 24218184 T^{7} + 317298898 T^{8} + 24218184 p T^{9} + 4008512 p^{2} T^{10} + 238032 p^{3} T^{11} + 38071 p^{4} T^{12} + 1470 p^{5} T^{13} + 257 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 296 T^{2} + 45660 T^{4} - 4991896 T^{6} + 415739654 T^{8} - 4991896 p^{2} T^{10} + 45660 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( ( 1 + 2 T + 277 T^{2} + 326 T^{3} + 31180 T^{4} + 326 p T^{5} + 277 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 83 | \( 1 - 282 T^{2} + 48733 T^{4} - 5918850 T^{6} + 560803488 T^{8} - 5918850 p^{2} T^{10} + 48733 p^{4} T^{12} - 282 p^{6} T^{14} + p^{8} T^{16} \) | |
| 89 | \( 1 - 18 T + 294 T^{2} - 3348 T^{3} + 32017 T^{4} - 3492 p T^{5} + 2819682 T^{6} - 344358 p T^{7} + 290866956 T^{8} - 344358 p^{2} T^{9} + 2819682 p^{2} T^{10} - 3492 p^{4} T^{11} + 32017 p^{4} T^{12} - 3348 p^{5} T^{13} + 294 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 54 T + 1634 T^{2} + 35748 T^{3} + 629470 T^{4} + 9407250 T^{5} + 122688872 T^{6} + 1420454826 T^{7} + 14743932787 T^{8} + 1420454826 p T^{9} + 122688872 p^{2} T^{10} + 9407250 p^{3} T^{11} + 629470 p^{4} T^{12} + 35748 p^{5} T^{13} + 1634 p^{6} T^{14} + 54 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.04969479874292474326173540571, −3.80738545438850390363077550142, −3.71027137868608999978408402436, −3.54162698824933471898446459937, −3.54134925062431481919316179599, −3.23990902169170392037671910448, −3.21325326088267648399852447054, −3.13681581693547915308390869092, −2.89452496836120799673419166325, −2.64980570845063449482869123277, −2.61991814842393730727331094433, −2.59372631858922707868771071369, −2.38585235550915691570208566319, −2.30616552449167365414273615638, −2.20237286051599991192325534386, −1.86602185291687175331336888165, −1.84333673423888684675837899759, −1.56985367691734936039870427237, −1.43512295046990089292425964970, −1.34799780408921185919779561263, −1.33471838791788171415647508572, −0.819170151948762855511968623689, −0.78667043121624930673587442428, −0.30060940435165579310297570843, −0.17660463204762279131216189463, 0.17660463204762279131216189463, 0.30060940435165579310297570843, 0.78667043121624930673587442428, 0.819170151948762855511968623689, 1.33471838791788171415647508572, 1.34799780408921185919779561263, 1.43512295046990089292425964970, 1.56985367691734936039870427237, 1.84333673423888684675837899759, 1.86602185291687175331336888165, 2.20237286051599991192325534386, 2.30616552449167365414273615638, 2.38585235550915691570208566319, 2.59372631858922707868771071369, 2.61991814842393730727331094433, 2.64980570845063449482869123277, 2.89452496836120799673419166325, 3.13681581693547915308390869092, 3.21325326088267648399852447054, 3.23990902169170392037671910448, 3.54134925062431481919316179599, 3.54162698824933471898446459937, 3.71027137868608999978408402436, 3.80738545438850390363077550142, 4.04969479874292474326173540571
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.