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Properties

Label 16-383e8-1.1-c0e8-0-0
Degree $16$
Conductor $4.630\times 10^{20}$
Sign $1$
Analytic cond. $1.78175\times 10^{-6}$
Root an. cond. $0.437197$
Motivic weight $0$
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  − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s − 19-s + 21-s − 23-s + 8·25-s − 29-s − 31-s + 34-s + 38-s − 42-s − 43-s + 46-s − 8·50-s + 51-s + 57-s + 58-s + 62-s − 67-s + 69-s − 71-s − 73-s − 8·75-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s − 19-s + 21-s − 23-s + 8·25-s − 29-s − 31-s + 34-s + 38-s − 42-s − 43-s + 46-s − 8·50-s + 51-s + 57-s + 58-s + 62-s − 67-s + 69-s − 71-s − 73-s − 8·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(383^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(383^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(383^{8}\)
Sign: $1$
Analytic conductor: \(1.78175\times 10^{-6}\)
Root analytic conductor: \(0.437197\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 383^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05906830282\)
\(L(\frac12)\) \(\approx\) \(0.05906830282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad383 \( ( 1 - T )^{8} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
5 \( ( 1 - T )^{8}( 1 + T )^{8} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
11 \( ( 1 - T )^{8}( 1 + T )^{8} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
19 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
29 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
31 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
37 \( ( 1 - T )^{8}( 1 + T )^{8} \)
41 \( ( 1 - T )^{8}( 1 + T )^{8} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
47 \( ( 1 - T )^{8}( 1 + T )^{8} \)
53 \( ( 1 - T )^{8}( 1 + T )^{8} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
71 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
73 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
79 \( ( 1 - T )^{8}( 1 + T )^{8} \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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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

−5.40607485090577472090485130689, −5.07762782650858468801127566999, −4.99906117642033026786865920718, −4.78449614205382577424308983650, −4.76114915807492017470986462553, −4.65324567835586820146965286077, −4.64750045437518672640831734230, −4.27157015197720836907545175175, −4.22476005282874989607483117241, −4.19840151935758864395708552915, −3.73813415167309892363009330846, −3.53136190238576533062351664112, −3.43966356762507517250240777169, −3.40525931811136009959446897213, −3.13829800942136174725975078055, −2.82194953247396000230178449300, −2.79003977329665825629135704491, −2.74998062063358052956518999714, −2.68118171349689725668581355238, −2.02511175885447331064263818073, −1.96306466555443317036835780579, −1.89063962636347752068405969415, −1.44306366962897073899379792133, −1.04980304949318996039678641634, −0.904508835895458011039725124394, 0.904508835895458011039725124394, 1.04980304949318996039678641634, 1.44306366962897073899379792133, 1.89063962636347752068405969415, 1.96306466555443317036835780579, 2.02511175885447331064263818073, 2.68118171349689725668581355238, 2.74998062063358052956518999714, 2.79003977329665825629135704491, 2.82194953247396000230178449300, 3.13829800942136174725975078055, 3.40525931811136009959446897213, 3.43966356762507517250240777169, 3.53136190238576533062351664112, 3.73813415167309892363009330846, 4.19840151935758864395708552915, 4.22476005282874989607483117241, 4.27157015197720836907545175175, 4.64750045437518672640831734230, 4.65324567835586820146965286077, 4.76114915807492017470986462553, 4.78449614205382577424308983650, 4.99906117642033026786865920718, 5.07762782650858468801127566999, 5.40607485090577472090485130689

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

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