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Properties

Label 16-543e8-1.1-c0e8-0-1
Degree $16$
Conductor $7.558\times 10^{21}$
Sign $1$
Analytic cond. $2.90841\times 10^{-5}$
Root an. cond. $0.520569$
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  − 3-s − 4-s + 9-s + 12-s + 2·13-s + 16-s − 2·25-s − 5·31-s − 36-s + 37-s − 2·39-s − 43-s − 48-s − 2·49-s − 2·52-s + 3·61-s + 3·67-s + 73-s + 2·75-s − 79-s + 5·93-s − 7·97-s + 2·100-s + 3·103-s + 3·109-s − 111-s + 2·117-s + ⋯
L(s)  = 1  − 3-s − 4-s + 9-s + 12-s + 2·13-s + 16-s − 2·25-s − 5·31-s − 36-s + 37-s − 2·39-s − 43-s − 48-s − 2·49-s − 2·52-s + 3·61-s + 3·67-s + 73-s + 2·75-s − 79-s + 5·93-s − 7·97-s + 2·100-s + 3·103-s + 3·109-s − 111-s + 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 181^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 181^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 181^{8}\)
Sign: $1$
Analytic conductor: \(2.90841\times 10^{-5}\)
Root analytic conductor: \(0.520569\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 181^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2005911446\)
\(L(\frac12)\) \(\approx\) \(0.2005911446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
181 \( ( 1 + T )^{8} \)
good2 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} )^{4} \)
19 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
23 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
41 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
47 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
53 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 + T )^{8}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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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

−5.05272366736118125494326206318, −4.88820387887531236561316388451, −4.76303594197575734933222071649, −4.64252265453481787733967606947, −4.39515282638115014656939077412, −4.23085825215494025011156341949, −4.18335368090535236945202858010, −4.00325670441245034974496699796, −3.93419256021990464741021265098, −3.61386178766256861467796823591, −3.59428431570073668161283522326, −3.53348907519064477534616874399, −3.45310221073286549396373781385, −3.38079184802935173772239902740, −3.00299973983605830480691325659, −2.78947007505498682778899569761, −2.67144502357362715685266851698, −2.21475974145462649748298355021, −2.14520056788820984482226556232, −1.92974809112542425622026434095, −1.87476621066490081003231714701, −1.69617294126856171640087948503, −1.35379066618973815338742509560, −1.01546015516420269048133602187, −0.888284891252456278448324172000, 0.888284891252456278448324172000, 1.01546015516420269048133602187, 1.35379066618973815338742509560, 1.69617294126856171640087948503, 1.87476621066490081003231714701, 1.92974809112542425622026434095, 2.14520056788820984482226556232, 2.21475974145462649748298355021, 2.67144502357362715685266851698, 2.78947007505498682778899569761, 3.00299973983605830480691325659, 3.38079184802935173772239902740, 3.45310221073286549396373781385, 3.53348907519064477534616874399, 3.59428431570073668161283522326, 3.61386178766256861467796823591, 3.93419256021990464741021265098, 4.00325670441245034974496699796, 4.18335368090535236945202858010, 4.23085825215494025011156341949, 4.39515282638115014656939077412, 4.64252265453481787733967606947, 4.76303594197575734933222071649, 4.88820387887531236561316388451, 5.05272366736118125494326206318

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

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