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Properties

Label 16-543e8-1.1-c0e8-0-1
Degree 1616
Conductor 7.558×10217.558\times 10^{21}
Sign 11
Analytic cond. 2.90841×10−52.90841\times 10^{-5}
Root an. cond. 0.5205690.520569
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

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

Λ(s)=((38⋅1818)s/2 ΓC(s)8 L(s)=( Λ(1−s)\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}
Λ(s)=((38⋅1818)s/2 ΓC(s)8 L(s)=( Λ(1−s)\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: 1616
Conductor: 38⋅18183^{8} \cdot 181^{8}
Sign: 11
Analytic conductor: 2.90841×10−52.90841\times 10^{-5}
Root analytic conductor: 0.5205690.520569
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 38⋅1818, ( :[0]8), 1)(16,\ 3^{8} \cdot 181^{8} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.20059114460.2005911446
L(12)L(\frac12) ≈\approx 0.20059114460.2005911446
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+T−T3−T4−T5+T7+T8 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8}
181 (1+T)8 ( 1 + T )^{8}
good2 1+T2−T6−T8−T10+T14+T16 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}
5 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
7 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
11 (1−T+T3−T4+T5−T7+T8)(1+T−T3−T4−T5+T7+T8) ( 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+T3−T4+T5−T7+T8)2 ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}
17 (1−T2+T4)4 ( 1 - T^{2} + T^{4} )^{4}
19 (1−T+T2)4(1+T+T2)4 ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}
23 1+T2−T6−T8−T10+T14+T16 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}
29 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
31 (1+T+T2)4(1+T−T3−T4−T5+T7+T8) ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )
37 (1−T+T2−T3+T4)2(1+T−T3−T4−T5+T7+T8) ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )
41 1+T2−T6−T8−T10+T14+T16 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}
43 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
47 1+T2−T6−T8−T10+T14+T16 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}
53 1+T2−T6−T8−T10+T14+T16 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}
59 (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}
61 (1−T+T2−T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
67 (1−T+T2)4(1+T−T3−T4−T5+T7+T8) ( 1 - T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )
71 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
73 (1−T+T2−T3+T4)2(1+T−T3−T4−T5+T7+T8) ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )
79 (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
83 1+T2−T6−T8−T10+T14+T16 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}
89 (1−T2+T4)4 ( 1 - T^{2} + T^{4} )^{4}
97 (1+T)8(1−T+T3−T4+T5−T7+T8) ( 1 + T )^{8}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(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 ZZ-function along the critical line

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