Properties

Label 16-384e8-1.1-c2e8-0-6
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $1.43658\times 10^{8}$
Root an. cond. $3.23469$
Motivic weight $2$
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  + 4·3-s − 8·7-s + 8·9-s − 24·19-s − 32·21-s + 80·25-s − 4·27-s + 56·31-s − 32·37-s + 136·43-s − 128·49-s − 96·57-s + 160·61-s − 64·63-s − 280·67-s − 80·73-s + 320·75-s + 408·79-s − 30·81-s + 224·93-s + 96·97-s − 488·103-s − 160·109-s − 128·111-s + 480·121-s + 127-s + 544·129-s + ⋯
L(s)  = 1  + 4/3·3-s − 8/7·7-s + 8/9·9-s − 1.26·19-s − 1.52·21-s + 16/5·25-s − 0.148·27-s + 1.80·31-s − 0.864·37-s + 3.16·43-s − 2.61·49-s − 1.68·57-s + 2.62·61-s − 1.01·63-s − 4.17·67-s − 1.09·73-s + 4.26·75-s + 5.16·79-s − 0.370·81-s + 2.40·93-s + 0.989·97-s − 4.73·103-s − 1.46·109-s − 1.15·111-s + 3.96·121-s + 0.00787·127-s + 4.21·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.43658\times 10^{8}\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(12.57599737\)
\(L(\frac12)\) \(\approx\) \(12.57599737\)
\(L(2)\) 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 \( 1 - 4 T + 8 T^{2} + 4 T^{3} - 22 p T^{4} + 4 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \)
good5 \( 1 - 16 p T^{2} + 3004 T^{4} - 18672 p T^{6} + 2634054 T^{8} - 18672 p^{5} T^{10} + 3004 p^{8} T^{12} - 16 p^{13} T^{14} + p^{16} T^{16} \)
7 \( ( 1 + 4 T + 88 T^{2} + 220 T^{3} + 3646 T^{4} + 220 p^{2} T^{5} + 88 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
11 \( 1 - 480 T^{2} + 125212 T^{4} - 22531104 T^{6} + 3078264198 T^{8} - 22531104 p^{4} T^{10} + 125212 p^{8} T^{12} - 480 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 + 316 T^{2} - 256 T^{3} + 57510 T^{4} - 256 p^{2} T^{5} + 316 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( 1 - 1064 T^{2} + 654556 T^{4} - 282654744 T^{6} + 92440824774 T^{8} - 282654744 p^{4} T^{10} + 654556 p^{8} T^{12} - 1064 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 12 T + 664 T^{2} - 4588 T^{3} + 120414 T^{4} - 4588 p^{2} T^{5} + 664 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 - 1896 T^{2} + 2093020 T^{4} - 1543350744 T^{6} + 914302437702 T^{8} - 1543350744 p^{4} T^{10} + 2093020 p^{8} T^{12} - 1896 p^{12} T^{14} + p^{16} T^{16} \)
29 \( 1 - 3024 T^{2} + 4188220 T^{4} - 3460574000 T^{6} + 2571955018182 T^{8} - 3460574000 p^{4} T^{10} + 4188220 p^{8} T^{12} - 3024 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 28 T + 3016 T^{2} - 63204 T^{3} + 3999966 T^{4} - 63204 p^{2} T^{5} + 3016 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 16 T + 1948 T^{2} - 25232 T^{3} + 1096870 T^{4} - 25232 p^{2} T^{5} + 1948 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 2472 T^{2} + 3015004 T^{4} - 2001918872 T^{6} + 1122734027334 T^{8} - 2001918872 p^{4} T^{10} + 3015004 p^{8} T^{12} - 2472 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 - 68 T + 6808 T^{2} - 284700 T^{3} + 17420958 T^{4} - 284700 p^{2} T^{5} + 6808 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 12296 T^{2} + 72283420 T^{4} - 268133584440 T^{6} + 697058148522822 T^{8} - 268133584440 p^{4} T^{10} + 72283420 p^{8} T^{12} - 12296 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 304 T^{2} + 25898428 T^{4} + 6393839056 T^{6} + 288293861465158 T^{8} + 6393839056 p^{4} T^{10} + 25898428 p^{8} T^{12} + 304 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 - 19008 T^{2} + 181109212 T^{4} - 1092163345344 T^{6} + 4537129454477574 T^{8} - 1092163345344 p^{4} T^{10} + 181109212 p^{8} T^{12} - 19008 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 80 T + 12700 T^{2} - 630832 T^{3} + 62237030 T^{4} - 630832 p^{2} T^{5} + 12700 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 140 T + 18856 T^{2} + 1550068 T^{3} + 122202110 T^{4} + 1550068 p^{2} T^{5} + 18856 p^{4} T^{6} + 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 - 17640 T^{2} + 102948316 T^{4} - 54971476568 T^{6} - 1448379553007034 T^{8} - 54971476568 p^{4} T^{10} + 102948316 p^{8} T^{12} - 17640 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 40 T + 15772 T^{2} + 446872 T^{3} + 110402182 T^{4} + 446872 p^{2} T^{5} + 15772 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - 204 T + 32488 T^{2} - 3143092 T^{3} + 288852126 T^{4} - 3143092 p^{2} T^{5} + 32488 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( 1 - 41696 T^{2} + 833769628 T^{4} - 10294236337952 T^{6} + 85511640874023430 T^{8} - 10294236337952 p^{4} T^{10} + 833769628 p^{8} T^{12} - 41696 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 17736 T^{2} + 318384412 T^{4} - 3335315895288 T^{6} + 31854516109881798 T^{8} - 3335315895288 p^{4} T^{10} + 318384412 p^{8} T^{12} - 17736 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 - 48 T + 29500 T^{2} - 1080784 T^{3} + 386488710 T^{4} - 1080784 p^{2} T^{5} + 29500 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−4.72187496390366878594770890058, −4.71210281140747890797516423120, −4.47956539728406419654575353720, −4.30068384479366205265991068582, −4.20880378904729459461227640765, −3.96985038582627626565472470330, −3.79539159312775555207300583753, −3.79040558303223486085321330655, −3.35889445733072195196147726481, −3.32207178578772582375903866233, −3.26488069102010351935595793005, −3.12681041232824466861589936992, −3.00459792155750266339613860987, −2.53895069861475347375307809329, −2.53648298752366689640721377140, −2.48586382215703987785121047212, −2.35579461991162504429855341847, −2.13288167011758992012355743752, −1.76052064855141286203243805785, −1.40788260227451033975989848682, −1.35698355360052007025516023476, −1.24387185107191800511290707845, −0.61284391498372858234643930832, −0.47177555516465047279712054696, −0.45706459300081457804752250188, 0.45706459300081457804752250188, 0.47177555516465047279712054696, 0.61284391498372858234643930832, 1.24387185107191800511290707845, 1.35698355360052007025516023476, 1.40788260227451033975989848682, 1.76052064855141286203243805785, 2.13288167011758992012355743752, 2.35579461991162504429855341847, 2.48586382215703987785121047212, 2.53648298752366689640721377140, 2.53895069861475347375307809329, 3.00459792155750266339613860987, 3.12681041232824466861589936992, 3.26488069102010351935595793005, 3.32207178578772582375903866233, 3.35889445733072195196147726481, 3.79040558303223486085321330655, 3.79539159312775555207300583753, 3.96985038582627626565472470330, 4.20880378904729459461227640765, 4.30068384479366205265991068582, 4.47956539728406419654575353720, 4.71210281140747890797516423120, 4.72187496390366878594770890058

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

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