Properties

Label 16-363e8-1.1-c2e8-0-9
Degree $16$
Conductor $3.015\times 10^{20}$
Sign $1$
Analytic cond. $9.16080\times 10^{7}$
Root an. cond. $3.14500$
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  + 5·3-s + 3·4-s + 28·7-s + 18·9-s + 15·12-s + 44·13-s − 25·16-s + 68·19-s + 140·21-s + 121·25-s + 40·27-s + 84·28-s + 2·31-s + 54·36-s + 140·37-s + 220·39-s − 78·43-s − 125·48-s + 126·49-s + 132·52-s + 340·57-s + 22·61-s + 504·63-s − 102·64-s + 184·67-s − 378·73-s + 605·75-s + ⋯
L(s)  = 1  + 5/3·3-s + 3/4·4-s + 4·7-s + 2·9-s + 5/4·12-s + 3.38·13-s − 1.56·16-s + 3.57·19-s + 20/3·21-s + 4.83·25-s + 1.48·27-s + 3·28-s + 2/31·31-s + 3/2·36-s + 3.78·37-s + 5.64·39-s − 1.81·43-s − 2.60·48-s + 18/7·49-s + 2.53·52-s + 5.96·57-s + 0.360·61-s + 8·63-s − 1.59·64-s + 2.74·67-s − 5.17·73-s + 8.06·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(9.16080\times 10^{7}\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 11^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(116.5760784\)
\(L(\frac12)\) \(\approx\) \(116.5760784\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5 T + 7 T^{2} + 5 p T^{3} - 8 p^{2} T^{4} + 5 p^{3} T^{5} + 7 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \)
11 \( 1 \)
good2 \( 1 - 3 T^{2} + 17 p T^{4} - 75 T^{6} + 715 T^{8} - 75 p^{4} T^{10} + 17 p^{9} T^{12} - 3 p^{12} T^{14} + p^{16} T^{16} \)
5 \( 1 - 121 T^{2} + 7337 T^{4} - 296499 T^{6} + 8656516 T^{8} - 296499 p^{4} T^{10} + 7337 p^{8} T^{12} - 121 p^{12} T^{14} + p^{16} T^{16} \)
7 \( ( 1 - 2 p T + 33 p T^{2} - 1920 T^{3} + 17440 T^{4} - 1920 p^{2} T^{5} + 33 p^{5} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 22 T + 591 T^{2} - 656 p T^{3} + 147080 T^{4} - 656 p^{3} T^{5} + 591 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( 1 - 1578 T^{2} + 1238593 T^{4} - 618939546 T^{6} + 213084965335 T^{8} - 618939546 p^{4} T^{10} + 1238593 p^{8} T^{12} - 1578 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 - 34 T + 1451 T^{2} - 28128 T^{3} + 37999 p T^{4} - 28128 p^{2} T^{5} + 1451 p^{4} T^{6} - 34 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 - 2384 T^{2} + 3158579 T^{4} - 2740573584 T^{6} + 1706195561800 T^{8} - 2740573584 p^{4} T^{10} + 3158579 p^{8} T^{12} - 2384 p^{12} T^{14} + p^{16} T^{16} \)
29 \( 1 - 6544 T^{2} + 18882716 T^{4} - 31382514288 T^{6} + 32859171662086 T^{8} - 31382514288 p^{4} T^{10} + 18882716 p^{8} T^{12} - 6544 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - T + 2135 T^{2} - 33669 T^{3} + 2159224 T^{4} - 33669 p^{2} T^{5} + 2135 p^{4} T^{6} - p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 70 T + 4365 T^{2} - 179810 T^{3} + 7505012 T^{4} - 179810 p^{2} T^{5} + 4365 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 4015 T^{2} + 11867438 T^{4} - 27037614090 T^{6} + 49932045282823 T^{8} - 27037614090 p^{4} T^{10} + 11867438 p^{8} T^{12} - 4015 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 39 T + 5868 T^{2} + 143114 T^{3} + 14177853 T^{4} + 143114 p^{2} T^{5} + 5868 p^{4} T^{6} + 39 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 6435 T^{2} + 600479 p T^{4} - 87544651725 T^{6} + 217222507276588 T^{8} - 87544651725 p^{4} T^{10} + 600479 p^{9} T^{12} - 6435 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 - 16839 T^{2} + 137076745 T^{4} - 690522689481 T^{6} + 2339818923445564 T^{8} - 690522689481 p^{4} T^{10} + 137076745 p^{8} T^{12} - 16839 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 - 4212 T^{2} + 7920259 T^{4} - 10638195330 T^{6} + 106243371880855 T^{8} - 10638195330 p^{4} T^{10} + 7920259 p^{8} T^{12} - 4212 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 11 T + 10335 T^{2} - 185079 T^{3} + 50401864 T^{4} - 185079 p^{2} T^{5} + 10335 p^{4} T^{6} - 11 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 - 92 T + 14029 T^{2} - 686396 T^{3} + 73672501 T^{4} - 686396 p^{2} T^{5} + 14029 p^{4} T^{6} - 92 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 - 37101 T^{2} + 8685579 p T^{4} - 5998803572371 T^{6} + 37352176354902420 T^{8} - 5998803572371 p^{4} T^{10} + 8685579 p^{9} T^{12} - 37101 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 189 T + 33673 T^{2} + 3338049 T^{3} + 304426708 T^{4} + 3338049 p^{2} T^{5} + 33673 p^{4} T^{6} + 189 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 126 T + 21061 T^{2} + 1430202 T^{3} + 157715956 T^{4} + 1430202 p^{2} T^{5} + 21061 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( 1 - 50355 T^{2} + 1136844414 T^{4} - 15049164252490 T^{6} + 127863558175452231 T^{8} - 15049164252490 p^{4} T^{10} + 1136844414 p^{8} T^{12} - 50355 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 54839 T^{2} + 1370536886 T^{4} - 20394246717258 T^{6} + 197742529962478711 T^{8} - 20394246717258 p^{4} T^{10} + 1370536886 p^{8} T^{12} - 54839 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 + 162 T + 44929 T^{2} + 4591386 T^{3} + 665405191 T^{4} + 4591386 p^{2} T^{5} + 44929 p^{4} T^{6} + 162 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−4.80120482569513839545016727621, −4.66656792438108857843492540725, −4.64115410227152727973559106240, −4.32647844953056788902623661266, −4.29657862250094771026318062361, −4.08942042470303363911223621763, −4.00184727532026384677513303579, −3.71735900695389851761510610049, −3.59251502854917619735712835356, −3.52782114546494455855929767098, −3.00656895534658703299893137475, −2.88244643573784140699139865527, −2.84529142656888042313113446736, −2.78727044434312311386649069651, −2.74871458133415072275991495083, −2.74526104422325158996378519729, −2.08549562349983896742448902270, −1.73088376690319417451339637060, −1.55916667345525612497982650865, −1.49043015313875821702785336318, −1.43625806949560011018261247788, −1.36100807542284497127266131234, −1.27433738315227101277751307437, −0.66715295302583485843994878347, −0.64371247556165195330443826624, 0.64371247556165195330443826624, 0.66715295302583485843994878347, 1.27433738315227101277751307437, 1.36100807542284497127266131234, 1.43625806949560011018261247788, 1.49043015313875821702785336318, 1.55916667345525612497982650865, 1.73088376690319417451339637060, 2.08549562349983896742448902270, 2.74526104422325158996378519729, 2.74871458133415072275991495083, 2.78727044434312311386649069651, 2.84529142656888042313113446736, 2.88244643573784140699139865527, 3.00656895534658703299893137475, 3.52782114546494455855929767098, 3.59251502854917619735712835356, 3.71735900695389851761510610049, 4.00184727532026384677513303579, 4.08942042470303363911223621763, 4.29657862250094771026318062361, 4.32647844953056788902623661266, 4.64115410227152727973559106240, 4.66656792438108857843492540725, 4.80120482569513839545016727621

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

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