Properties

Label 16-150e8-1.1-c1e8-0-1
Degree $16$
Conductor $2.563\times 10^{17}$
Sign $1$
Analytic cond. $4.23591$
Root an. cond. $1.09442$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 9-s + 10·11-s − 20·13-s + 10·17-s − 8·19-s − 10·23-s − 5·25-s − 22·29-s + 24·31-s + 36-s − 20·37-s + 22·41-s + 10·44-s + 10·47-s + 32·49-s − 20·52-s − 30·53-s − 20·59-s + 10·67-s + 10·68-s + 20·71-s − 20·73-s − 8·76-s + 16·79-s + 70·83-s − 34·89-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s + 3.01·11-s − 5.54·13-s + 2.42·17-s − 1.83·19-s − 2.08·23-s − 25-s − 4.08·29-s + 4.31·31-s + 1/6·36-s − 3.28·37-s + 3.43·41-s + 1.50·44-s + 1.45·47-s + 32/7·49-s − 2.77·52-s − 4.12·53-s − 2.60·59-s + 1.22·67-s + 1.21·68-s + 2.37·71-s − 2.34·73-s − 0.917·76-s + 1.80·79-s + 7.68·83-s − 3.60·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.23591\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.141758339\)
\(L(\frac12)\) \(\approx\) \(1.141758339\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 + p T^{2} + 2 p T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{8} \)
good7 \( 1 - 32 T^{2} + 500 T^{4} - 736 p T^{6} + 40294 T^{8} - 736 p^{3} T^{10} + 500 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 10 T + 28 T^{2} + 60 T^{3} - 617 T^{4} + 1790 T^{5} - 794 T^{6} - 18400 T^{7} + 94725 T^{8} - 18400 p T^{9} - 794 p^{2} T^{10} + 1790 p^{3} T^{11} - 617 p^{4} T^{12} + 60 p^{5} T^{13} + 28 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 20 T + 202 T^{2} + 1420 T^{3} + 8095 T^{4} + 40660 T^{5} + 14284 p T^{6} + 772160 T^{7} + 2918789 T^{8} + 772160 p T^{9} + 14284 p^{3} T^{10} + 40660 p^{3} T^{11} + 8095 p^{4} T^{12} + 1420 p^{5} T^{13} + 202 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 10 T + 78 T^{2} - 590 T^{3} + 3615 T^{4} - 20330 T^{5} + 105768 T^{6} - 488740 T^{7} + 2089449 T^{8} - 488740 p T^{9} + 105768 p^{2} T^{10} - 20330 p^{3} T^{11} + 3615 p^{4} T^{12} - 590 p^{5} T^{13} + 78 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 8 T - 10 T^{2} - 270 T^{3} - 565 T^{4} + 5224 T^{5} + 28422 T^{6} - 2940 p T^{7} - 41505 p T^{8} - 2940 p^{2} T^{9} + 28422 p^{2} T^{10} + 5224 p^{3} T^{11} - 565 p^{4} T^{12} - 270 p^{5} T^{13} - 10 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 + 10 T + 116 T^{2} + 950 T^{3} + 6927 T^{4} + 45910 T^{5} + 11616 p T^{6} + 1460600 T^{7} + 7269605 T^{8} + 1460600 p T^{9} + 11616 p^{3} T^{10} + 45910 p^{3} T^{11} + 6927 p^{4} T^{12} + 950 p^{5} T^{13} + 116 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 22 T + 10 p T^{2} + 2930 T^{3} + 24395 T^{4} + 175526 T^{5} + 1131792 T^{6} + 6698140 T^{7} + 37055745 T^{8} + 6698140 p T^{9} + 1131792 p^{2} T^{10} + 175526 p^{3} T^{11} + 24395 p^{4} T^{12} + 2930 p^{5} T^{13} + 10 p^{7} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 24 T + 230 T^{2} - 990 T^{3} - 685 T^{4} + 32568 T^{5} - 115202 T^{6} - 726180 T^{7} + 8308045 T^{8} - 726180 p T^{9} - 115202 p^{2} T^{10} + 32568 p^{3} T^{11} - 685 p^{4} T^{12} - 990 p^{5} T^{13} + 230 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 20 T + 275 T^{2} + 3140 T^{3} + 29831 T^{4} + 251620 T^{5} + 1934025 T^{6} + 13441540 T^{7} + 85392936 T^{8} + 13441540 p T^{9} + 1934025 p^{2} T^{10} + 251620 p^{3} T^{11} + 29831 p^{4} T^{12} + 3140 p^{5} T^{13} + 275 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 22 T + 146 T^{2} + 142 T^{3} - 6829 T^{4} + 52634 T^{5} - 236776 T^{6} - 967444 T^{7} + 17720057 T^{8} - 967444 p T^{9} - 236776 p^{2} T^{10} + 52634 p^{3} T^{11} - 6829 p^{4} T^{12} + 142 p^{5} T^{13} + 146 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 120 T^{2} + 10856 T^{4} - 676920 T^{6} + 33149566 T^{8} - 676920 p^{2} T^{10} + 10856 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 10 T + 164 T^{2} - 1060 T^{3} + 6767 T^{4} - 10550 T^{5} - 301398 T^{6} + 3376520 T^{7} - 34256075 T^{8} + 3376520 p T^{9} - 301398 p^{2} T^{10} - 10550 p^{3} T^{11} + 6767 p^{4} T^{12} - 1060 p^{5} T^{13} + 164 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 30 T + 425 T^{2} + 3090 T^{3} + 5031 T^{4} - 120570 T^{5} - 1120625 T^{6} - 3254310 T^{7} + 1334576 T^{8} - 3254310 p T^{9} - 1120625 p^{2} T^{10} - 120570 p^{3} T^{11} + 5031 p^{4} T^{12} + 3090 p^{5} T^{13} + 425 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 20 T + 242 T^{2} + 2760 T^{3} + 31483 T^{4} + 335220 T^{5} + 3074684 T^{6} + 25371200 T^{7} + 201486805 T^{8} + 25371200 p T^{9} + 3074684 p^{2} T^{10} + 335220 p^{3} T^{11} + 31483 p^{4} T^{12} + 2760 p^{5} T^{13} + 242 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 118 T^{2} + 10203 T^{4} + 835676 T^{6} + 64821605 T^{8} + 835676 p^{2} T^{10} + 10203 p^{4} T^{12} + 118 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 10 T + 88 T^{2} - 1660 T^{3} + 11615 T^{4} - 86790 T^{5} + 911158 T^{6} - 5793400 T^{7} + 42525189 T^{8} - 5793400 p T^{9} + 911158 p^{2} T^{10} - 86790 p^{3} T^{11} + 11615 p^{4} T^{12} - 1660 p^{5} T^{13} + 88 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 20 T + 118 T^{2} + 490 T^{3} - 14397 T^{4} + 149580 T^{5} - 402414 T^{6} - 9696700 T^{7} + 137916525 T^{8} - 9696700 p T^{9} - 402414 p^{2} T^{10} + 149580 p^{3} T^{11} - 14397 p^{4} T^{12} + 490 p^{5} T^{13} + 118 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 20 T + 282 T^{2} + 3780 T^{3} + 46315 T^{4} + 505780 T^{5} + 5254892 T^{6} + 48564720 T^{7} + 416447549 T^{8} + 48564720 p T^{9} + 5254892 p^{2} T^{10} + 505780 p^{3} T^{11} + 46315 p^{4} T^{12} + 3780 p^{5} T^{13} + 282 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 16 T + 274 T^{2} - 44 p T^{3} + 48131 T^{4} - 506768 T^{5} + 5479676 T^{6} - 51684448 T^{7} + 496487797 T^{8} - 51684448 p T^{9} + 5479676 p^{2} T^{10} - 506768 p^{3} T^{11} + 48131 p^{4} T^{12} - 44 p^{6} T^{13} + 274 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 70 T + 2432 T^{2} - 56730 T^{3} + 1010935 T^{4} - 14768130 T^{5} + 184247072 T^{6} - 2010015520 T^{7} + 19419043189 T^{8} - 2010015520 p T^{9} + 184247072 p^{2} T^{10} - 14768130 p^{3} T^{11} + 1010935 p^{4} T^{12} - 56730 p^{5} T^{13} + 2432 p^{6} T^{14} - 70 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 34 T + 409 T^{2} + 1014 T^{3} - 27449 T^{4} - 344318 T^{5} - 919769 T^{6} + 22713862 T^{7} + 350578752 T^{8} + 22713862 p T^{9} - 919769 p^{2} T^{10} - 344318 p^{3} T^{11} - 27449 p^{4} T^{12} + 1014 p^{5} T^{13} + 409 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 60 T + 1990 T^{2} - 47940 T^{3} + 920391 T^{4} - 14754060 T^{5} + 202723580 T^{6} - 2424996600 T^{7} + 25463108381 T^{8} - 2424996600 p T^{9} + 202723580 p^{2} T^{10} - 14754060 p^{3} T^{11} + 920391 p^{4} T^{12} - 47940 p^{5} T^{13} + 1990 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \)
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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

−5.93704100825626943491352389076, −5.91963445896214015498339546110, −5.88737954755091702272459861193, −5.59586891974056645186885715717, −5.30440925796893097663807519466, −5.05356691651534727532203987559, −5.03272960524734459525352367324, −4.96820418358940770032079564638, −4.74200744321745670760205748768, −4.45537292895232149990941377331, −4.37543552111195535176490240179, −4.04057303394289109748380676273, −3.92387088891517629959245578434, −3.82274750758155501537675217923, −3.70197360395045360126036730490, −3.50128620220071149324657511328, −3.21801842446514913325743962387, −2.62218181372433609251246945968, −2.52240532732498267869034480613, −2.49655488322092396469502388147, −2.23124935863865476961635814660, −2.10304965717315225729640861085, −1.56973763982750033986901123134, −1.49307621749145247293830725356, −0.56250169283673559377675110514, 0.56250169283673559377675110514, 1.49307621749145247293830725356, 1.56973763982750033986901123134, 2.10304965717315225729640861085, 2.23124935863865476961635814660, 2.49655488322092396469502388147, 2.52240532732498267869034480613, 2.62218181372433609251246945968, 3.21801842446514913325743962387, 3.50128620220071149324657511328, 3.70197360395045360126036730490, 3.82274750758155501537675217923, 3.92387088891517629959245578434, 4.04057303394289109748380676273, 4.37543552111195535176490240179, 4.45537292895232149990941377331, 4.74200744321745670760205748768, 4.96820418358940770032079564638, 5.03272960524734459525352367324, 5.05356691651534727532203987559, 5.30440925796893097663807519466, 5.59586891974056645186885715717, 5.88737954755091702272459861193, 5.91963445896214015498339546110, 5.93704100825626943491352389076

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

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