Properties

Label 16-21e16-1.1-c3e8-0-0
Degree $16$
Conductor $1.431\times 10^{21}$
Sign $1$
Analytic cond. $2.10105\times 10^{11}$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 34·8-s + 100·11-s + 44·16-s + 200·22-s + 352·23-s + 314·25-s − 520·29-s + 78·32-s − 212·37-s + 1.08e3·43-s + 100·44-s + 704·46-s + 628·50-s + 16·53-s − 1.04e3·58-s + 371·64-s + 1.94e3·67-s − 4.49e3·71-s − 424·74-s + 1.04e3·79-s + 2.16e3·86-s − 3.40e3·88-s + 352·92-s + 314·100-s + 32·106-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/8·4-s − 1.50·8-s + 2.74·11-s + 0.687·16-s + 1.93·22-s + 3.19·23-s + 2.51·25-s − 3.32·29-s + 0.430·32-s − 0.941·37-s + 3.83·43-s + 0.342·44-s + 2.25·46-s + 1.77·50-s + 0.0414·53-s − 2.35·58-s + 0.724·64-s + 3.54·67-s − 7.51·71-s − 0.666·74-s + 1.49·79-s + 2.70·86-s − 4.11·88-s + 0.398·92-s + 0.313·100-s + 0.0293·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.10105\times 10^{11}\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.314023444\)
\(L(\frac12)\) \(\approx\) \(1.314023444\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - T + T^{2} + p^{4} T^{3} - 9 p^{3} T^{4} + p^{7} T^{5} + p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} )^{2} \)
5 \( 1 - 314 T^{2} + 50562 T^{4} - 5270176 T^{6} + 522562031 T^{8} - 5270176 p^{6} T^{10} + 50562 p^{12} T^{12} - 314 p^{18} T^{14} + p^{24} T^{16} \)
11 \( ( 1 - 50 T - 202 T^{2} - 2000 T^{3} + 2201743 T^{4} - 2000 p^{3} T^{5} - 202 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
13 \( ( 1 + 5554 T^{2} + 17209282 T^{4} + 5554 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
17 \( 1 - 5088 T^{2} + 14421310 T^{4} + 187282685952 T^{6} - 1076010335607741 T^{8} + 187282685952 p^{6} T^{10} + 14421310 p^{12} T^{12} - 5088 p^{18} T^{14} + p^{24} T^{16} \)
19 \( 1 - 12690 T^{2} + 28454938 T^{4} - 488430486000 T^{6} + 7788364400899983 T^{8} - 488430486000 p^{6} T^{10} + 28454938 p^{12} T^{12} - 12690 p^{18} T^{14} + p^{24} T^{16} \)
23 \( ( 1 - 176 T - 62 T^{2} - 1179904 T^{3} + 438436563 T^{4} - 1179904 p^{3} T^{5} - 62 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
29 \( ( 1 + 130 T + 49818 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
31 \( 1 - 19060 T^{2} + 373250298 T^{4} + 34021605583600 T^{6} - 1109978980966458157 T^{8} + 34021605583600 p^{6} T^{10} + 373250298 p^{12} T^{12} - 19060 p^{18} T^{14} + p^{24} T^{16} \)
37 \( ( 1 + 106 T - 92294 T^{2} + 235744 T^{3} + 7583597383 T^{4} + 235744 p^{3} T^{5} - 92294 p^{6} T^{6} + 106 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
41 \( ( 1 + 208848 T^{2} + 19595539298 T^{4} + 208848 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
43 \( ( 1 - 270 T + 97614 T^{2} - 270 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
47 \( 1 - 228052 T^{2} + 17880021370 T^{4} - 2866445491787152 T^{6} + \)\(48\!\cdots\!99\)\( T^{8} - 2866445491787152 p^{6} T^{10} + 17880021370 p^{12} T^{12} - 228052 p^{18} T^{14} + p^{24} T^{16} \)
53 \( ( 1 - 8 T - 276646 T^{2} + 168352 T^{3} + 54394534843 T^{4} + 168352 p^{3} T^{5} - 276646 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
59 \( 1 - 632162 T^{2} + 222723302586 T^{4} - 58503068402380912 T^{6} + \)\(12\!\cdots\!79\)\( T^{8} - 58503068402380912 p^{6} T^{10} + 222723302586 p^{12} T^{12} - 632162 p^{18} T^{14} + p^{24} T^{16} \)
61 \( 1 - 8978 p T^{2} + 149331839266 T^{4} - 426964025450528 p T^{6} + \)\(45\!\cdots\!79\)\( T^{8} - 426964025450528 p^{7} T^{10} + 149331839266 p^{12} T^{12} - 8978 p^{19} T^{14} + p^{24} T^{16} \)
67 \( ( 1 - 972 T + 200922 T^{2} - 138350592 T^{3} + 178716222683 T^{4} - 138350592 p^{3} T^{5} + 200922 p^{6} T^{6} - 972 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
71 \( ( 1 + 1124 T + 1018926 T^{2} + 1124 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
73 \( 1 - 1279312 T^{2} + 931531176990 T^{4} - 514845763213402112 T^{6} + \)\(22\!\cdots\!79\)\( T^{8} - 514845763213402112 p^{6} T^{10} + 931531176990 p^{12} T^{12} - 1279312 p^{18} T^{14} + p^{24} T^{16} \)
79 \( ( 1 - 524 T - 205806 T^{2} + 264984704 T^{3} - 147697266061 T^{4} + 264984704 p^{3} T^{5} - 205806 p^{6} T^{6} - 524 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
83 \( ( 1 + 2275682 T^{2} + 1948536513034 T^{4} + 2275682 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
89 \( 1 + 804000 T^{2} - 70725036962 T^{4} - 222564522147840000 T^{6} - \)\(53\!\cdots\!77\)\( T^{8} - 222564522147840000 p^{6} T^{10} - 70725036962 p^{12} T^{12} + 804000 p^{18} T^{14} + p^{24} T^{16} \)
97 \( ( 1 + 567808 T^{2} + 211724872834 T^{4} + 567808 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
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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

−4.38267656264164180617006595768, −4.35377466797817829458240878885, −4.17590967825588022483092569755, −3.94573188379906561813527270681, −3.60026566391656209111885081886, −3.59984745803966536901229991466, −3.59801557699201928618041007033, −3.42126856380859622562557685897, −3.39844787839411077451351292559, −3.10121476898386212165224015051, −3.01520101441856137733242150906, −2.83999427450375027508499936254, −2.61851756776793985591799656771, −2.53054546334356427722579398318, −2.20594695440074016721310206189, −1.97177779649744958670490088452, −1.88290798214248062184794840050, −1.71605578157271568870840825735, −1.24810100100459303977182759757, −1.21410128171113276454091241884, −1.02849059037218215720234536147, −0.999143529491332731475522291361, −0.60347384408802594537617989148, −0.60270185352272281651558128632, −0.05112971679106043644064919278, 0.05112971679106043644064919278, 0.60270185352272281651558128632, 0.60347384408802594537617989148, 0.999143529491332731475522291361, 1.02849059037218215720234536147, 1.21410128171113276454091241884, 1.24810100100459303977182759757, 1.71605578157271568870840825735, 1.88290798214248062184794840050, 1.97177779649744958670490088452, 2.20594695440074016721310206189, 2.53054546334356427722579398318, 2.61851756776793985591799656771, 2.83999427450375027508499936254, 3.01520101441856137733242150906, 3.10121476898386212165224015051, 3.39844787839411077451351292559, 3.42126856380859622562557685897, 3.59801557699201928618041007033, 3.59984745803966536901229991466, 3.60026566391656209111885081886, 3.94573188379906561813527270681, 4.17590967825588022483092569755, 4.35377466797817829458240878885, 4.38267656264164180617006595768

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

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