Properties

Label 16-1520e8-1.1-c1e8-0-0
Degree $16$
Conductor $2.849\times 10^{25}$
Sign $1$
Analytic cond. $4.70939\times 10^{8}$
Root an. cond. $3.48385$
Motivic weight $1$
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  + 3·3-s − 4·5-s + 8·7-s + 10·9-s + 4·11-s − 7·13-s − 12·15-s + 17-s − 5·19-s + 24·21-s + 2·23-s + 6·25-s + 19·27-s + 29-s + 12·33-s − 32·35-s − 4·37-s − 21·39-s + 8·41-s + 43-s − 40·45-s − 12·47-s − 6·49-s + 3·51-s + 5·53-s − 16·55-s − 15·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.78·5-s + 3.02·7-s + 10/3·9-s + 1.20·11-s − 1.94·13-s − 3.09·15-s + 0.242·17-s − 1.14·19-s + 5.23·21-s + 0.417·23-s + 6/5·25-s + 3.65·27-s + 0.185·29-s + 2.08·33-s − 5.40·35-s − 0.657·37-s − 3.36·39-s + 1.24·41-s + 0.152·43-s − 5.96·45-s − 1.75·47-s − 6/7·49-s + 0.420·51-s + 0.686·53-s − 2.15·55-s − 1.98·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(4.70939\times 10^{8}\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7988785877\)
\(L(\frac12)\) \(\approx\) \(0.7988785877\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + T + T^{2} )^{4} \)
19 \( 1 + 5 T + 31 T^{2} + 67 T^{3} + 395 T^{4} + 67 p T^{5} + 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
good3 \( 1 - p T - T^{2} + 14 T^{3} - 10 T^{4} - 22 T^{5} + 26 T^{6} + T^{7} - 11 T^{8} + p T^{9} + 26 p^{2} T^{10} - 22 p^{3} T^{11} - 10 p^{4} T^{12} + 14 p^{5} T^{13} - p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \)
7 \( ( 1 - 4 T + 27 T^{2} - 69 T^{3} + 272 T^{4} - 69 p T^{5} + 27 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 2 T + 19 T^{2} - 47 T^{3} + 179 T^{4} - 47 p T^{5} + 19 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 7 T - 10 T^{2} - 87 T^{3} + 551 T^{4} + 1480 T^{5} - 9928 T^{6} - 3324 T^{7} + 178356 T^{8} - 3324 p T^{9} - 9928 p^{2} T^{10} + 1480 p^{3} T^{11} + 551 p^{4} T^{12} - 87 p^{5} T^{13} - 10 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - T - 37 T^{2} + 98 T^{3} + 554 T^{4} - 2096 T^{5} - 5804 T^{6} + 945 p T^{7} + 106065 T^{8} + 945 p^{2} T^{9} - 5804 p^{2} T^{10} - 2096 p^{3} T^{11} + 554 p^{4} T^{12} + 98 p^{5} T^{13} - 37 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 2 T - 71 T^{2} + 140 T^{3} + 2937 T^{4} - 4618 T^{5} - 86145 T^{6} + 48856 T^{7} + 2108699 T^{8} + 48856 p T^{9} - 86145 p^{2} T^{10} - 4618 p^{3} T^{11} + 2937 p^{4} T^{12} + 140 p^{5} T^{13} - 71 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - T - 52 T^{2} + 509 T^{3} + 1277 T^{4} - 20540 T^{5} + 93655 T^{6} + 443301 T^{7} - 3745740 T^{8} + 443301 p T^{9} + 93655 p^{2} T^{10} - 20540 p^{3} T^{11} + 1277 p^{4} T^{12} + 509 p^{5} T^{13} - 52 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 + 57 T^{2} + 5 T^{3} + 2675 T^{4} + 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 2 T + 117 T^{2} + 99 T^{3} + 5802 T^{4} + 99 p T^{5} + 117 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 8 T - 13 T^{2} - 590 T^{3} + 5111 T^{4} + 9890 T^{5} + 199801 T^{6} - 1514064 T^{7} - 3467655 T^{8} - 1514064 p T^{9} + 199801 p^{2} T^{10} + 9890 p^{3} T^{11} + 5111 p^{4} T^{12} - 590 p^{5} T^{13} - 13 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - T - 73 T^{2} - 210 T^{3} + 2084 T^{4} + 14876 T^{5} + 25160 T^{6} - 413805 T^{7} - 1612143 T^{8} - 413805 p T^{9} + 25160 p^{2} T^{10} + 14876 p^{3} T^{11} + 2084 p^{4} T^{12} - 210 p^{5} T^{13} - 73 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 12 T + 11 T^{2} + 70 T^{3} + 2765 T^{4} - 1092 T^{5} - 27713 T^{6} - 366602 T^{7} - 8145777 T^{8} - 366602 p T^{9} - 27713 p^{2} T^{10} - 1092 p^{3} T^{11} + 2765 p^{4} T^{12} + 70 p^{5} T^{13} + 11 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 5 T - 101 T^{2} + 506 T^{3} + 5352 T^{4} - 24832 T^{5} - 88212 T^{6} + 707641 T^{7} - 3097531 T^{8} + 707641 p T^{9} - 88212 p^{2} T^{10} - 24832 p^{3} T^{11} + 5352 p^{4} T^{12} + 506 p^{5} T^{13} - 101 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 5 T - 86 T^{2} - 215 T^{3} + 2385 T^{4} - 14660 T^{5} - 81789 T^{6} + 1000355 T^{7} + 10693034 T^{8} + 1000355 p T^{9} - 81789 p^{2} T^{10} - 14660 p^{3} T^{11} + 2385 p^{4} T^{12} - 215 p^{5} T^{13} - 86 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 114 T^{2} + 176 T^{3} + 3481 T^{4} - 15400 T^{5} - 228578 T^{6} + 386584 T^{7} + 27858884 T^{8} + 386584 p T^{9} - 228578 p^{2} T^{10} - 15400 p^{3} T^{11} + 3481 p^{4} T^{12} + 176 p^{5} T^{13} - 114 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 4 T - 216 T^{2} + 728 T^{3} + 28338 T^{4} - 70180 T^{5} - 2597792 T^{6} + 2011948 T^{7} + 193609507 T^{8} + 2011948 p T^{9} - 2597792 p^{2} T^{10} - 70180 p^{3} T^{11} + 28338 p^{4} T^{12} + 728 p^{5} T^{13} - 216 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 20 T + 25 T^{2} + 830 T^{3} + 14400 T^{4} - 140590 T^{5} - 883200 T^{6} - 1962095 T^{7} + 162929144 T^{8} - 1962095 p T^{9} - 883200 p^{2} T^{10} - 140590 p^{3} T^{11} + 14400 p^{4} T^{12} + 830 p^{5} T^{13} + 25 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 20 T + 13 T^{2} + 582 T^{3} + 22431 T^{4} - 149340 T^{5} - 1710159 T^{6} + 34288 p T^{7} + 167178049 T^{8} + 34288 p^{2} T^{9} - 1710159 p^{2} T^{10} - 149340 p^{3} T^{11} + 22431 p^{4} T^{12} + 582 p^{5} T^{13} + 13 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 17 T - 99 T^{2} + 18 p T^{3} + 33753 T^{4} - 218511 T^{5} - 3340534 T^{6} + 1230695 T^{7} + 402442002 T^{8} + 1230695 p T^{9} - 3340534 p^{2} T^{10} - 218511 p^{3} T^{11} + 33753 p^{4} T^{12} + 18 p^{6} T^{13} - 99 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \)
83 \( ( 1 + T + 270 T^{2} + 194 T^{3} + 31408 T^{4} + 194 p T^{5} + 270 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 11 T - 145 T^{2} - 172 T^{3} + 26027 T^{4} - 59285 T^{5} - 2268638 T^{6} + 481263 T^{7} + 100591392 T^{8} + 481263 p T^{9} - 2268638 p^{2} T^{10} - 59285 p^{3} T^{11} + 26027 p^{4} T^{12} - 172 p^{5} T^{13} - 145 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + T - 121 T^{2} - 2070 T^{3} + 1496 T^{4} + 215440 T^{5} + 1891076 T^{6} - 10407525 T^{7} - 192172179 T^{8} - 10407525 p T^{9} + 1891076 p^{2} T^{10} + 215440 p^{3} T^{11} + 1496 p^{4} T^{12} - 2070 p^{5} T^{13} - 121 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.95791297018816450298980362253, −3.85100646275465312001260682401, −3.85001691503578083364876978780, −3.66719494208723822687389965638, −3.61350813700661900686264140956, −3.50566300119912289913359451543, −3.24765723229919047175921915060, −3.06911508356420823020047129651, −2.99122302212572584689176181222, −2.92881321296041894219250331758, −2.61955652257050509348914854238, −2.47548948774566073190943349525, −2.43889713321150768884480966694, −2.20810986168350107341368007094, −2.18170088007259169205682767218, −1.93953733543096489774511647582, −1.84562970725537258309089945938, −1.65415628602824726986864591345, −1.55082738815426603998228757966, −1.24939105551554254026306767424, −1.14819192795897110374669973161, −1.13822038520596102667790051142, −1.05840784890103537444358374895, −0.31348889387892447092718346522, −0.079465659315318079634297236963, 0.079465659315318079634297236963, 0.31348889387892447092718346522, 1.05840784890103537444358374895, 1.13822038520596102667790051142, 1.14819192795897110374669973161, 1.24939105551554254026306767424, 1.55082738815426603998228757966, 1.65415628602824726986864591345, 1.84562970725537258309089945938, 1.93953733543096489774511647582, 2.18170088007259169205682767218, 2.20810986168350107341368007094, 2.43889713321150768884480966694, 2.47548948774566073190943349525, 2.61955652257050509348914854238, 2.92881321296041894219250331758, 2.99122302212572584689176181222, 3.06911508356420823020047129651, 3.24765723229919047175921915060, 3.50566300119912289913359451543, 3.61350813700661900686264140956, 3.66719494208723822687389965638, 3.85001691503578083364876978780, 3.85100646275465312001260682401, 3.95791297018816450298980362253

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

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