Properties

Label 16-7e16-1.1-c5e8-0-0
Degree $16$
Conductor $3.323\times 10^{13}$
Sign $1$
Analytic cond. $1.45496\times 10^{7}$
Root an. cond. $2.80335$
Motivic weight $5$
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  + 10·2-s + 109·4-s + 790·8-s + 376·9-s + 1.95e3·11-s + 5.91e3·16-s + 3.76e3·18-s + 1.95e4·22-s + 7.13e3·23-s + 4.86e3·25-s − 6.70e3·29-s + 2.91e4·32-s + 4.09e4·36-s + 9.20e3·37-s + 4.08e4·43-s + 2.12e5·44-s + 7.13e4·46-s + 4.86e4·50-s + 1.02e5·53-s − 6.70e4·58-s + 1.54e5·64-s + 2.28e4·67-s − 3.07e5·71-s + 2.97e5·72-s + 9.20e4·74-s + 9.06e4·79-s + 8.83e4·81-s + ⋯
L(s)  = 1  + 1.76·2-s + 3.40·4-s + 4.36·8-s + 1.54·9-s + 4.86·11-s + 5.77·16-s + 2.73·18-s + 8.59·22-s + 2.81·23-s + 1.55·25-s − 1.48·29-s + 5.02·32-s + 5.27·36-s + 1.10·37-s + 3.37·43-s + 16.5·44-s + 4.97·46-s + 2.75·50-s + 5.03·53-s − 2.61·58-s + 4.72·64-s + 0.623·67-s − 7.24·71-s + 6.75·72-s + 1.95·74-s + 1.63·79-s + 1.49·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16}\)
Sign: $1$
Analytic conductor: \(1.45496\times 10^{7}\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(141.4569029\)
\(L(\frac12)\) \(\approx\) \(141.4569029\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( ( 1 - 5 T - 17 T^{2} + 55 p T^{3} - 15 p^{2} T^{4} + 55 p^{6} T^{5} - 17 p^{10} T^{6} - 5 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
3 \( 1 - 376 T^{2} + 17674 p T^{4} + 11183744 T^{6} - 4028466509 T^{8} + 11183744 p^{10} T^{10} + 17674 p^{21} T^{12} - 376 p^{30} T^{14} + p^{40} T^{16} \)
5 \( 1 - 4868 T^{2} + 1134618 T^{4} - 14757614608 T^{6} + 182292626703011 T^{8} - 14757614608 p^{10} T^{10} + 1134618 p^{20} T^{12} - 4868 p^{30} T^{14} + p^{40} T^{16} \)
11 \( ( 1 - 976 T + 396398 T^{2} - 228458176 T^{3} + 129983225707 T^{4} - 228458176 p^{5} T^{5} + 396398 p^{10} T^{6} - 976 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
13 \( ( 1 + 224500 T^{2} - 32848089674 T^{4} + 224500 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
17 \( 1 - 3065760 T^{2} + 3024788435134 T^{4} - 7180341456093511680 T^{6} + \)\(17\!\cdots\!55\)\( T^{8} - 7180341456093511680 p^{10} T^{10} + 3024788435134 p^{20} T^{12} - 3065760 p^{30} T^{14} + p^{40} T^{16} \)
19 \( 1 - 8544504 T^{2} + 42956548559710 T^{4} - \)\(15\!\cdots\!16\)\( T^{6} + \)\(41\!\cdots\!19\)\( T^{8} - \)\(15\!\cdots\!16\)\( p^{10} T^{10} + 42956548559710 p^{20} T^{12} - 8544504 p^{30} T^{14} + p^{40} T^{16} \)
23 \( ( 1 - 3568 T + 18274 T^{2} + 572078848 T^{3} + 38238668640819 T^{4} + 572078848 p^{5} T^{5} + 18274 p^{10} T^{6} - 3568 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
29 \( ( 1 + 1676 T + 24360510 T^{2} + 1676 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
31 \( 1 - 29395756 T^{2} + 205881674857002 T^{4} + \)\(28\!\cdots\!08\)\( T^{6} - \)\(94\!\cdots\!81\)\( T^{8} + \)\(28\!\cdots\!08\)\( p^{10} T^{10} + 205881674857002 p^{20} T^{12} - 29395756 p^{30} T^{14} + p^{40} T^{16} \)
37 \( ( 1 - 4604 T - 122725214 T^{2} - 24097870064 T^{3} + 14435095249662139 T^{4} - 24097870064 p^{5} T^{5} - 122725214 p^{10} T^{6} - 4604 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
41 \( ( 1 + 212331360 T^{2} + 36724946155085474 T^{4} + 212331360 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
43 \( ( 1 - 10224 T + 293018130 T^{2} - 10224 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
47 \( 1 - 568199404 T^{2} + 150307861516914922 T^{4} - \)\(38\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!51\)\( T^{8} - \)\(38\!\cdots\!84\)\( p^{10} T^{10} + 150307861516914922 p^{20} T^{12} - 568199404 p^{30} T^{14} + p^{40} T^{16} \)
53 \( ( 1 - 51460 T + 1339504322 T^{2} - 24301279586320 T^{3} + 430181182369297435 T^{4} - 24301279586320 p^{5} T^{5} + 1339504322 p^{10} T^{6} - 51460 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
59 \( 1 - 1609944152 T^{2} + 1200185847501479934 T^{4} - \)\(59\!\cdots\!36\)\( T^{6} + \)\(35\!\cdots\!95\)\( T^{8} - \)\(59\!\cdots\!36\)\( p^{10} T^{10} + 1200185847501479934 p^{20} T^{12} - 1609944152 p^{30} T^{14} + p^{40} T^{16} \)
61 \( 1 - 1094138468 T^{2} - 412207137807081734 T^{4} - \)\(19\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!39\)\( T^{8} - \)\(19\!\cdots\!08\)\( p^{10} T^{10} - 412207137807081734 p^{20} T^{12} - 1094138468 p^{30} T^{14} + p^{40} T^{16} \)
67 \( ( 1 - 11448 T + 1220364042 T^{2} + 43382854855296 T^{3} - 813190274442182533 T^{4} + 43382854855296 p^{5} T^{5} + 1220364042 p^{10} T^{6} - 11448 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
71 \( ( 1 + 76912 T + 4562060670 T^{2} + 76912 p^{5} T^{3} + p^{10} T^{4} )^{4} \)
73 \( 1 - 2170565248 T^{2} + 4286181938510097630 T^{4} + \)\(17\!\cdots\!52\)\( T^{6} - \)\(38\!\cdots\!21\)\( T^{8} + \)\(17\!\cdots\!52\)\( p^{10} T^{10} + 4286181938510097630 p^{20} T^{12} - 2170565248 p^{30} T^{14} + p^{40} T^{16} \)
79 \( ( 1 - 45344 T - 4098421134 T^{2} - 17533255168 T^{3} + 22082918642969361635 T^{4} - 17533255168 p^{5} T^{5} - 4098421134 p^{10} T^{6} - 45344 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
83 \( ( 1 + 6721792472 T^{2} + 21939991980485194594 T^{4} + 6721792472 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
89 \( 1 - 14718300768 T^{2} + \)\(10\!\cdots\!66\)\( T^{4} - \)\(78\!\cdots\!08\)\( T^{6} + \)\(56\!\cdots\!39\)\( T^{8} - \)\(78\!\cdots\!08\)\( p^{10} T^{10} + \)\(10\!\cdots\!66\)\( p^{20} T^{12} - 14718300768 p^{30} T^{14} + p^{40} T^{16} \)
97 \( ( 1 + 22489794400 T^{2} + 25388976518139394 p^{2} T^{4} + 22489794400 p^{10} T^{6} + p^{20} T^{8} )^{2} \)
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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

−6.42138441784189415185922489682, −6.01915244244738253358265111134, −6.01900307272834309488203636124, −5.87360222460017352265734089575, −5.68517510031743784433705286655, −5.57541669947735751893826178191, −5.18780835087480420964670398745, −4.70268449039605136858083475364, −4.60010617073679749296822578628, −4.55618893221208174406971048281, −4.34602994532188341267153832990, −3.90959184547142977400297893082, −3.86833127191890956139710806789, −3.65959453145992713367181972330, −3.55882501981732478680781781255, −3.28543427976210476182246684436, −2.75957390372344750956776190908, −2.59258709503750789230589184514, −2.35230236436993423393757883629, −2.04075490907682545513325787181, −1.40806333170032811630673517089, −1.32334256562072071445122266764, −1.24743851160453769340770646466, −1.03031907024905714493777001066, −0.65574159389265924212460772692, 0.65574159389265924212460772692, 1.03031907024905714493777001066, 1.24743851160453769340770646466, 1.32334256562072071445122266764, 1.40806333170032811630673517089, 2.04075490907682545513325787181, 2.35230236436993423393757883629, 2.59258709503750789230589184514, 2.75957390372344750956776190908, 3.28543427976210476182246684436, 3.55882501981732478680781781255, 3.65959453145992713367181972330, 3.86833127191890956139710806789, 3.90959184547142977400297893082, 4.34602994532188341267153832990, 4.55618893221208174406971048281, 4.60010617073679749296822578628, 4.70268449039605136858083475364, 5.18780835087480420964670398745, 5.57541669947735751893826178191, 5.68517510031743784433705286655, 5.87360222460017352265734089575, 6.01900307272834309488203636124, 6.01915244244738253358265111134, 6.42138441784189415185922489682

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

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