Properties

Label 16-1512e8-1.1-c1e8-0-4
Degree $16$
Conductor $2.732\times 10^{25}$
Sign $1$
Analytic cond. $4.51472\times 10^{8}$
Root an. cond. $3.47467$
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  − 4·5-s − 4·7-s + 6·11-s − 3·13-s + 16·17-s − 4·19-s + 5·23-s + 11·25-s − 29-s + 11·31-s + 16·35-s + 54·37-s − 2·41-s − 11·43-s − 7·47-s + 6·49-s + 8·53-s − 24·55-s − 9·59-s − 7·61-s + 12·65-s − 12·67-s + 24·71-s + 26·73-s − 24·77-s − 22·79-s + 6·83-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.51·7-s + 1.80·11-s − 0.832·13-s + 3.88·17-s − 0.917·19-s + 1.04·23-s + 11/5·25-s − 0.185·29-s + 1.97·31-s + 2.70·35-s + 8.87·37-s − 0.312·41-s − 1.67·43-s − 1.02·47-s + 6/7·49-s + 1.09·53-s − 3.23·55-s − 1.17·59-s − 0.896·61-s + 1.48·65-s − 1.46·67-s + 2.84·71-s + 3.04·73-s − 2.73·77-s − 2.47·79-s + 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(4.51472\times 10^{8}\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.109942168\)
\(L(\frac12)\) \(\approx\) \(3.109942168\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T + T^{2} )^{4} \)
good5 \( 1 + 4 T + p T^{2} - 18 T^{3} - 94 T^{4} - 232 T^{5} - 22 p T^{6} + 1011 T^{7} + 3826 T^{8} + 1011 p T^{9} - 22 p^{3} T^{10} - 232 p^{3} T^{11} - 94 p^{4} T^{12} - 18 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 6 T + T^{2} + 24 T^{3} + 49 T^{4} + 306 T^{5} - 2153 T^{6} + 1236 T^{7} + 7063 T^{8} + 1236 p T^{9} - 2153 p^{2} T^{10} + 306 p^{3} T^{11} + 49 p^{4} T^{12} + 24 p^{5} T^{13} + p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + 337 T^{4} - 720 T^{5} + 2222 T^{6} + 906 p T^{7} - 3392 p T^{8} + 906 p^{2} T^{9} + 2222 p^{2} T^{10} - 720 p^{3} T^{11} + 337 p^{4} T^{12} + 3 p^{6} T^{13} - 16 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 - 8 T + 35 T^{2} - 167 T^{3} + 886 T^{4} - 167 p T^{5} + 35 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 2 T + 55 T^{2} + 41 T^{3} + 1309 T^{4} + 41 p T^{5} + 55 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 5 T - 28 T^{2} + 177 T^{3} - 25 T^{4} + 494 T^{5} - 12137 T^{6} - 44331 T^{7} + 708646 T^{8} - 44331 p T^{9} - 12137 p^{2} T^{10} + 494 p^{3} T^{11} - 25 p^{4} T^{12} + 177 p^{5} T^{13} - 28 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + T - 49 T^{2} - 294 T^{3} + 1136 T^{4} + 11204 T^{5} + 49096 T^{6} - 227337 T^{7} - 2198075 T^{8} - 227337 p T^{9} + 49096 p^{2} T^{10} + 11204 p^{3} T^{11} + 1136 p^{4} T^{12} - 294 p^{5} T^{13} - 49 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 11 T - 39 T^{2} + 356 T^{3} + 5954 T^{4} - 25902 T^{5} - 215174 T^{6} + 3835 T^{7} + 11112081 T^{8} + 3835 p T^{9} - 215174 p^{2} T^{10} - 25902 p^{3} T^{11} + 5954 p^{4} T^{12} + 356 p^{5} T^{13} - 39 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 4104 p T^{5} + 412 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 + 2 T - 37 T^{2} - 432 T^{3} - 841 T^{4} + 13042 T^{5} + 100489 T^{6} - 72294 T^{7} - 3776099 T^{8} - 72294 p T^{9} + 100489 p^{2} T^{10} + 13042 p^{3} T^{11} - 841 p^{4} T^{12} - 432 p^{5} T^{13} - 37 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 11 T - 27 T^{2} - 596 T^{3} + 140 T^{4} + 2688 T^{5} - 152864 T^{6} + 330797 T^{7} + 13009029 T^{8} + 330797 p T^{9} - 152864 p^{2} T^{10} + 2688 p^{3} T^{11} + 140 p^{4} T^{12} - 596 p^{5} T^{13} - 27 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 7 T - 61 T^{2} + 174 T^{3} + 5108 T^{4} - 21898 T^{5} - 62918 T^{6} + 808359 T^{7} - 1538879 T^{8} + 808359 p T^{9} - 62918 p^{2} T^{10} - 21898 p^{3} T^{11} + 5108 p^{4} T^{12} + 174 p^{5} T^{13} - 61 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
53 \( ( 1 - 4 T + 83 T^{2} + 197 T^{3} + 2722 T^{4} + 197 p T^{5} + 83 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 9 T - 119 T^{2} - 1116 T^{3} + 10336 T^{4} + 79866 T^{5} - 564434 T^{6} - 2176353 T^{7} + 30131725 T^{8} - 2176353 p T^{9} - 564434 p^{2} T^{10} + 79866 p^{3} T^{11} + 10336 p^{4} T^{12} - 1116 p^{5} T^{13} - 119 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 7 T - 144 T^{2} - 1315 T^{3} + 11783 T^{4} + 112692 T^{5} - 470051 T^{6} - 3262925 T^{7} + 21742764 T^{8} - 3262925 p T^{9} - 470051 p^{2} T^{10} + 112692 p^{3} T^{11} + 11783 p^{4} T^{12} - 1315 p^{5} T^{13} - 144 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 12 T - 139 T^{2} - 1338 T^{3} + 22729 T^{4} + 133506 T^{5} - 1954153 T^{6} - 2368722 T^{7} + 170018875 T^{8} - 2368722 p T^{9} - 1954153 p^{2} T^{10} + 133506 p^{3} T^{11} + 22729 p^{4} T^{12} - 1338 p^{5} T^{13} - 139 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 - 12 T + 167 T^{2} - 591 T^{3} + 7587 T^{4} - 591 p T^{5} + 167 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 1504 p T^{5} + 232 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 + 22 T + 3 p T^{2} + 1616 T^{3} + 2882 T^{4} - 87342 T^{5} - 574370 T^{6} + 1050079 T^{7} + 29800944 T^{8} + 1050079 p T^{9} - 574370 p^{2} T^{10} - 87342 p^{3} T^{11} + 2882 p^{4} T^{12} + 1616 p^{5} T^{13} + 3 p^{7} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 6 T - 113 T^{2} + 1704 T^{3} + 3151 T^{4} - 166296 T^{5} + 1416277 T^{6} + 7918434 T^{7} - 172589093 T^{8} + 7918434 p T^{9} + 1416277 p^{2} T^{10} - 166296 p^{3} T^{11} + 3151 p^{4} T^{12} + 1704 p^{5} T^{13} - 113 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 - 14 T + 323 T^{2} - 2963 T^{3} + 41152 T^{4} - 2963 p T^{5} + 323 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + T - 123 T^{2} - 856 T^{3} - 4138 T^{4} + 70242 T^{5} + 167026 T^{6} - 386291 T^{7} + 107246157 T^{8} - 386291 p T^{9} + 167026 p^{2} T^{10} + 70242 p^{3} T^{11} - 4138 p^{4} T^{12} - 856 p^{5} T^{13} - 123 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.04522361250130862551672504798, −3.87571030154558673604408157414, −3.75535425068320062576182688949, −3.64905913344268348597026754699, −3.34633857047542129479924384110, −3.27752032399634203550236798517, −3.24137943801185898614230691495, −3.22568286629598781079894164794, −3.05675881666105406293369725137, −2.94308262656663552567665990149, −2.82587186032380850409672705213, −2.74523920777087289644766563658, −2.40652514341558369044263041208, −2.32272371405387009023909993638, −2.20435442010935546895192074024, −2.00709352066068120863824962433, −1.97148874326865594975696996189, −1.37815566710146801242040531434, −1.34273902263531724511882366959, −1.20574431645130938130159315764, −0.954440240043490572087085975888, −0.809447279753490207202708999874, −0.78599749546763811495601008518, −0.72027068800574698837961844984, −0.16979773887953815343510923958, 0.16979773887953815343510923958, 0.72027068800574698837961844984, 0.78599749546763811495601008518, 0.809447279753490207202708999874, 0.954440240043490572087085975888, 1.20574431645130938130159315764, 1.34273902263531724511882366959, 1.37815566710146801242040531434, 1.97148874326865594975696996189, 2.00709352066068120863824962433, 2.20435442010935546895192074024, 2.32272371405387009023909993638, 2.40652514341558369044263041208, 2.74523920777087289644766563658, 2.82587186032380850409672705213, 2.94308262656663552567665990149, 3.05675881666105406293369725137, 3.22568286629598781079894164794, 3.24137943801185898614230691495, 3.27752032399634203550236798517, 3.34633857047542129479924384110, 3.64905913344268348597026754699, 3.75535425068320062576182688949, 3.87571030154558673604408157414, 4.04522361250130862551672504798

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

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