Properties

Label 16-392e8-1.1-c2e8-0-2
Degree $16$
Conductor $5.576\times 10^{20}$
Sign $1$
Analytic cond. $1.69421\times 10^{8}$
Root an. cond. $3.26821$
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  − 8·3-s + 6·4-s + 56·9-s − 16·11-s − 48·12-s + 16·16-s − 72·17-s − 8·19-s − 16·25-s − 368·27-s + 128·33-s + 336·36-s − 80·41-s − 160·43-s − 96·44-s − 128·48-s + 576·51-s + 64·57-s − 184·59-s + 72·64-s + 224·67-s − 432·68-s − 232·73-s + 128·75-s − 48·76-s + 1.93e3·81-s + 176·83-s + ⋯
L(s)  = 1  − 8/3·3-s + 3/2·4-s + 56/9·9-s − 1.45·11-s − 4·12-s + 16-s − 4.23·17-s − 0.421·19-s − 0.639·25-s − 13.6·27-s + 3.87·33-s + 28/3·36-s − 1.95·41-s − 3.72·43-s − 2.18·44-s − 8/3·48-s + 11.2·51-s + 1.12·57-s − 3.11·59-s + 9/8·64-s + 3.34·67-s − 6.35·68-s − 3.17·73-s + 1.70·75-s − 0.631·76-s + 23.9·81-s + 2.12·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.69421\times 10^{8}\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.04152988923\)
\(L(\frac12)\) \(\approx\) \(0.04152988923\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 3 p T^{2} + 5 p^{2} T^{4} - 3 p^{5} T^{6} + p^{8} T^{8} \)
7 \( 1 \)
good3 \( ( 1 + 4 T - 4 T^{2} + 8 T^{3} + 175 T^{4} + 8 p^{2} T^{5} - 4 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
5 \( 1 + 16 T^{2} + 102 p T^{4} - 24064 T^{6} - 486109 T^{8} - 24064 p^{4} T^{10} + 102 p^{9} T^{12} + 16 p^{12} T^{14} + p^{16} T^{16} \)
11 \( ( 1 + 8 T - 122 T^{2} - 448 T^{3} + 12211 T^{4} - 448 p^{2} T^{5} - 122 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 592 T^{2} + 143170 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 36 T + 426 T^{2} + 10512 T^{3} + 298835 T^{4} + 10512 p^{2} T^{5} + 426 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 4 T + 12 T^{2} - 2872 T^{3} - 136081 T^{4} - 2872 p^{2} T^{5} + 12 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 + 268 T^{2} - 405462 T^{4} - 22082128 T^{6} + 129391640531 T^{8} - 22082128 p^{4} T^{10} - 405462 p^{8} T^{12} + 268 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 1178 T^{2} + p^{4} T^{4} )^{4} \)
31 \( 1 + 1380 T^{2} + 484426 T^{4} - 589353840 T^{6} - 595318269165 T^{8} - 589353840 p^{4} T^{10} + 484426 p^{8} T^{12} + 1380 p^{12} T^{14} + p^{16} T^{16} \)
37 \( 1 + 1780 T^{2} + 1136778 T^{4} - 3055726000 T^{6} - 5323083215437 T^{8} - 3055726000 p^{4} T^{10} + 1136778 p^{8} T^{12} + 1780 p^{12} T^{14} + p^{16} T^{16} \)
41 \( ( 1 + 20 T + 3174 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
43 \( ( 1 + 40 T + 4090 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( 1 + 7492 T^{2} + 32739594 T^{4} + 102124261136 T^{6} + 249405993329555 T^{8} + 102124261136 p^{4} T^{10} + 32739594 p^{8} T^{12} + 7492 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 1604 T^{2} + 8727850 T^{4} - 35185337584 T^{6} - 44676002634701 T^{8} - 35185337584 p^{4} T^{10} + 8727850 p^{8} T^{12} + 1604 p^{12} T^{14} + p^{16} T^{16} \)
59 \( ( 1 + 92 T - 372 T^{2} + 172408 T^{3} + 36494351 T^{4} + 172408 p^{2} T^{5} - 372 p^{4} T^{6} + 92 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( 1 + 13232 T^{2} + 103975486 T^{4} + 574515656192 T^{6} + 2440762377018979 T^{8} + 574515656192 p^{4} T^{10} + 103975486 p^{8} T^{12} + 13232 p^{12} T^{14} + p^{16} T^{16} \)
67 \( ( 1 - 112 T + 942 T^{2} - 293888 T^{3} + 58145267 T^{4} - 293888 p^{2} T^{5} + 942 p^{4} T^{6} - 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 + 116 T - 6 p T^{2} + 375376 T^{3} + 92937971 T^{4} + 375376 p^{2} T^{5} - 6 p^{5} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 + 16900 T^{2} + 142729866 T^{4} + 1098161526800 T^{6} + 7773091247541395 T^{8} + 1098161526800 p^{4} T^{10} + 142729866 p^{8} T^{12} + 16900 p^{12} T^{14} + p^{16} T^{16} \)
83 \( ( 1 - 44 T + 13924 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
89 \( ( 1 + 156 T + 3562 T^{2} + 769392 T^{3} + 176051379 T^{4} + 769392 p^{2} T^{5} + 3562 p^{4} T^{6} + 156 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 68 T + 3046 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−4.84842877330859044067170846259, −4.81522367254571877651332481371, −4.45455125392897318798270224884, −4.21681635257504838714362404663, −4.16613170307746630643215470744, −4.08524217481693482543230831404, −4.08360476411437487153847259271, −3.96833753522234535136425214854, −3.68866583163519922173914636842, −3.34295201545710756990606671114, −3.24957569346789214146818171524, −3.02627804151330172621115424146, −2.88507919816414874720348957065, −2.49116061879792985742416930648, −2.44962755747733803772544114420, −2.30795930888605462182185347613, −1.84509032184450168853680816366, −1.81644228363234411998547913021, −1.72802823498389380240309949772, −1.65164328733144217688609638787, −1.53160985749395844380932973073, −1.11646149995306018705732617102, −0.42900727732236826752626436687, −0.35199952416737476868592594143, −0.05601531877825411571668744695, 0.05601531877825411571668744695, 0.35199952416737476868592594143, 0.42900727732236826752626436687, 1.11646149995306018705732617102, 1.53160985749395844380932973073, 1.65164328733144217688609638787, 1.72802823498389380240309949772, 1.81644228363234411998547913021, 1.84509032184450168853680816366, 2.30795930888605462182185347613, 2.44962755747733803772544114420, 2.49116061879792985742416930648, 2.88507919816414874720348957065, 3.02627804151330172621115424146, 3.24957569346789214146818171524, 3.34295201545710756990606671114, 3.68866583163519922173914636842, 3.96833753522234535136425214854, 4.08360476411437487153847259271, 4.08524217481693482543230831404, 4.16613170307746630643215470744, 4.21681635257504838714362404663, 4.45455125392897318798270224884, 4.81522367254571877651332481371, 4.84842877330859044067170846259

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

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