↑

Properties

Label 16-1155e8-1.1-c0e8-0-1
Degree $16$
Conductor $3.167\times 10^{24}$
Sign $1$
Analytic cond. $0.0121874$
Root an. cond. $0.759223$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 9-s + 2·11-s + 16-s + 25-s − 4·29-s − 2·36-s − 4·44-s + 49-s + 2·99-s − 2·100-s + 8·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 2·176-s + 179-s + ⋯
L(s)  = 1  − 2·4-s + 9-s + 2·11-s + 16-s + 25-s − 4·29-s − 2·36-s − 4·44-s + 49-s + 2·99-s − 2·100-s + 8·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 2·176-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(0.0121874\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5212705359\)
\(L(\frac12)\) \(\approx\) \(0.5212705359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 + T^{2} )^{8} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 + T^{2} )^{8} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.22158401684550213104128494473, −4.13227608881315739835096999887, −4.12747945818839430585349040632, −4.11552504550977431232957800260, −4.01572193588355142139484238141, −4.01545344951715982630391463511, −3.97592554379337370475311465482, −3.44668285116196004010246749892, −3.31784775946854281687624922032, −3.30205462792196647919216545507, −3.18421409198057726799335671910, −3.17722489514966219404981179723, −2.87929122015353144949239620790, −2.82288620142241574497513372982, −2.47181757862547321923055113305, −2.31535240084069125657915237706, −2.06505044501560109794587399546, −1.97449069812931206431957072707, −1.81725985449129133163190639007, −1.78859598453094338331194961447, −1.52610180369650430253024447315, −1.39869966022524746553802161121, −1.00435722343690978763363889405, −0.883182437990402172295430772671, −0.57527468706362142150992287407, 0.57527468706362142150992287407, 0.883182437990402172295430772671, 1.00435722343690978763363889405, 1.39869966022524746553802161121, 1.52610180369650430253024447315, 1.78859598453094338331194961447, 1.81725985449129133163190639007, 1.97449069812931206431957072707, 2.06505044501560109794587399546, 2.31535240084069125657915237706, 2.47181757862547321923055113305, 2.82288620142241574497513372982, 2.87929122015353144949239620790, 3.17722489514966219404981179723, 3.18421409198057726799335671910, 3.30205462792196647919216545507, 3.31784775946854281687624922032, 3.44668285116196004010246749892, 3.97592554379337370475311465482, 4.01545344951715982630391463511, 4.01572193588355142139484238141, 4.11552504550977431232957800260, 4.12747945818839430585349040632, 4.13227608881315739835096999887, 4.22158401684550213104128494473

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

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