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Properties

Label 16-960e8-1.1-c2e8-0-11
Degree $16$
Conductor $7.214\times 10^{23}$
Sign $1$
Analytic cond. $2.19204\times 10^{11}$
Root an. cond. $5.11449$
Motivic weight $2$
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  − 12·9-s + 16·13-s − 16·17-s + 20·25-s + 96·29-s + 112·37-s + 112·41-s + 104·49-s − 224·53-s − 80·61-s + 272·73-s + 90·81-s − 48·89-s + 528·97-s + 416·101-s − 112·109-s − 176·113-s − 192·117-s + 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 192·153-s + 157-s + ⋯
L(s)  = 1  − 4/3·9-s + 1.23·13-s − 0.941·17-s + 4/5·25-s + 3.31·29-s + 3.02·37-s + 2.73·41-s + 2.12·49-s − 4.22·53-s − 1.31·61-s + 3.72·73-s + 10/9·81-s − 0.539·89-s + 5.44·97-s + 4.11·101-s − 1.02·109-s − 1.55·113-s − 1.64·117-s + 0.859·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.25·153-s + 0.00636·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.19204\times 10^{11}\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(13.85638325\)
\(L(\frac12)\) \(\approx\) \(13.85638325\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + p T^{2} )^{4} \)
5 \( ( 1 - p T^{2} )^{4} \)
good7 \( 1 - 104 T^{2} + 3804 T^{4} - 61144 T^{6} + 732230 T^{8} - 61144 p^{4} T^{10} + 3804 p^{8} T^{12} - 104 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 - 104 T^{2} + 25116 T^{4} + 998312 T^{6} + 87392006 T^{8} + 998312 p^{4} T^{10} + 25116 p^{8} T^{12} - 104 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 - 8 T + 396 T^{2} - 4792 T^{3} + 76550 T^{4} - 4792 p^{2} T^{5} + 396 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 8 T - 84 T^{2} - 3848 T^{3} - 35290 T^{4} - 3848 p^{2} T^{5} - 84 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( 1 - 200 T^{2} + 273628 T^{4} - 52427000 T^{6} + 42482298118 T^{8} - 52427000 p^{4} T^{10} + 273628 p^{8} T^{12} - 200 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 1544 T^{2} + 1318044 T^{4} - 803710264 T^{6} + 443101191110 T^{8} - 803710264 p^{4} T^{10} + 1318044 p^{8} T^{12} - 1544 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 24 T + 1646 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
31 \( 1 - 2824 T^{2} + 6208540 T^{4} - 8647788856 T^{6} + 9849609291334 T^{8} - 8647788856 p^{4} T^{10} + 6208540 p^{8} T^{12} - 2824 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 - 56 T + 3948 T^{2} - 157576 T^{3} + 7918598 T^{4} - 157576 p^{2} T^{5} + 3948 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 56 T + 2556 T^{2} - 74632 T^{3} + 3542726 T^{4} - 74632 p^{2} T^{5} + 2556 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 10760 T^{2} + 54445788 T^{4} - 171718842040 T^{6} + 375491195821958 T^{8} - 171718842040 p^{4} T^{10} + 54445788 p^{8} T^{12} - 10760 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 10888 T^{2} + 61255708 T^{4} - 225571130296 T^{6} + 586057028916550 T^{8} - 225571130296 p^{4} T^{10} + 61255708 p^{8} T^{12} - 10888 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 + 112 T + 13020 T^{2} + 887312 T^{3} + 57092774 T^{4} + 887312 p^{2} T^{5} + 13020 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 20456 T^{2} + 200671260 T^{4} - 1224887792344 T^{6} + 1465211341814 p^{2} T^{8} - 1224887792344 p^{4} T^{10} + 200671260 p^{8} T^{12} - 20456 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 40 T + 11964 T^{2} + 456920 T^{3} + 61872806 T^{4} + 456920 p^{2} T^{5} + 11964 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 4744 T^{2} + 570460 T^{4} + 34360533704 T^{6} - 13806137860346 T^{8} + 34360533704 p^{4} T^{10} + 570460 p^{8} T^{12} - 4744 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 19720 T^{2} + 218875804 T^{4} - 1700648520760 T^{6} + 9880226933129926 T^{8} - 1700648520760 p^{4} T^{10} + 218875804 p^{8} T^{12} - 19720 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 136 T + 19228 T^{2} - 1717816 T^{3} + 139076998 T^{4} - 1717816 p^{2} T^{5} + 19228 p^{4} T^{6} - 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 36104 T^{2} + 626466076 T^{4} - 6818196448568 T^{6} + 50837338178201926 T^{8} - 6818196448568 p^{4} T^{10} + 626466076 p^{8} T^{12} - 36104 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 29320 T^{2} + 467847004 T^{4} - 4970634792760 T^{6} + 39363161035770886 T^{8} - 4970634792760 p^{4} T^{10} + 467847004 p^{8} T^{12} - 29320 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 + 24 T + 4060 T^{2} - 454104 T^{3} - 10096506 T^{4} - 454104 p^{2} T^{5} + 4060 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 132 T + 20102 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−4.20264690686788907540007041479, −3.88393883557736479365470427748, −3.84871819377706010112035865623, −3.54310821417531300301322100590, −3.45995151985215168926365174377, −3.43457630346128456465185052877, −3.27062668539582058117309358556, −3.15726306005533341460631365865, −2.98299728832251637369083801920, −2.89027555929545871450418741713, −2.58767143801005855265787935310, −2.49104460086305845519706580189, −2.48108494750059151968467693901, −2.22115707467574197607466549765, −2.17471782505371670969584439017, −2.16780680631642085178827553249, −1.70438684797098199428260008406, −1.47685674435094951760060751857, −1.37959989915866487133945443283, −1.04871616666934553544796421107, −0.937341432871829972309011356238, −0.815599122447077493222529804736, −0.68537660877571038723214623289, −0.43065757070828910208124036501, −0.26917009535345646624705867534, 0.26917009535345646624705867534, 0.43065757070828910208124036501, 0.68537660877571038723214623289, 0.815599122447077493222529804736, 0.937341432871829972309011356238, 1.04871616666934553544796421107, 1.37959989915866487133945443283, 1.47685674435094951760060751857, 1.70438684797098199428260008406, 2.16780680631642085178827553249, 2.17471782505371670969584439017, 2.22115707467574197607466549765, 2.48108494750059151968467693901, 2.49104460086305845519706580189, 2.58767143801005855265787935310, 2.89027555929545871450418741713, 2.98299728832251637369083801920, 3.15726306005533341460631365865, 3.27062668539582058117309358556, 3.43457630346128456465185052877, 3.45995151985215168926365174377, 3.54310821417531300301322100590, 3.84871819377706010112035865623, 3.88393883557736479365470427748, 4.20264690686788907540007041479

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

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