# Properties

 Label 16-546e8-1.1-c1e8-0-12 Degree $16$ Conductor $7.898\times 10^{21}$ Sign $1$ Analytic cond. $130544.$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 4·3-s + 2·4-s + 6·9-s + 6·11-s + 8·12-s + 12·13-s + 16-s + 2·17-s − 12·19-s + 8·23-s + 8·25-s + 2·29-s + 24·33-s + 12·36-s + 18·37-s + 48·39-s + 12·41-s − 8·43-s + 12·44-s + 4·48-s + 2·49-s + 8·51-s + 24·52-s − 12·53-s − 48·57-s + 18·59-s − 8·61-s + ⋯
 L(s)  = 1 + 2.30·3-s + 4-s + 2·9-s + 1.80·11-s + 2.30·12-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 + 4.17·33-s + 2·36-s + 2.95·37-s + 7.68·39-s + 1.87·41-s − 1.21·43-s + 1.80·44-s + 0.577·48-s + 2/7·49-s + 1.12·51-s + 3.32·52-s − 1.64·53-s − 6.35·57-s + 2.34·59-s − 1.02·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \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^{8} \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: $$2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$130544.$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$27.36424772$$ $$L(\frac12)$$ $$\approx$$ $$27.36424772$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 - T^{2} + T^{4} )^{2}$$
3 $$( 1 - T + T^{2} )^{4}$$
7 $$( 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}$$
good5 $$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

−4.77292817459108798808381240618, −4.20994325428367096955048700307, −4.16483731699570470533197513691, −4.15570252696066319005374244641, −4.14250299671800697735016151326, −4.06099543942088371588117575318, −4.03818256803755228264191672854, −3.84351538409597409447509703236, −3.57077956221773455200562312936, −3.18616859703353666726230207951, −3.14060414376862508795376117042, −3.13249927276825195201384804925, −2.93915415698978448934695211243, −2.90985402050510826591347431082, −2.73310824370063157662183896347, −2.54695127516940982732816701043, −2.18757437735285719857719446244, −2.15183689530463916303899119836, −2.02637429201744097395803530814, −1.76186345992483143026681131053, −1.37603737750243930769747092059, −1.27720965018775831276361634425, −1.11541997439468983081592657501, −1.09057123630961569093770172167, −0.52292639274108066341199473540, 0.52292639274108066341199473540, 1.09057123630961569093770172167, 1.11541997439468983081592657501, 1.27720965018775831276361634425, 1.37603737750243930769747092059, 1.76186345992483143026681131053, 2.02637429201744097395803530814, 2.15183689530463916303899119836, 2.18757437735285719857719446244, 2.54695127516940982732816701043, 2.73310824370063157662183896347, 2.90985402050510826591347431082, 2.93915415698978448934695211243, 3.13249927276825195201384804925, 3.14060414376862508795376117042, 3.18616859703353666726230207951, 3.57077956221773455200562312936, 3.84351538409597409447509703236, 4.03818256803755228264191672854, 4.06099543942088371588117575318, 4.14250299671800697735016151326, 4.15570252696066319005374244641, 4.16483731699570470533197513691, 4.20994325428367096955048700307, 4.77292817459108798808381240618

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.