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Properties

Label 16-33e16-1.1-c0e8-0-2
Degree 1616
Conductor 1.978×10241.978\times 10^{24}
Sign 11
Analytic cond. 0.007611670.00761167
Root an. cond. 0.7372120.737212
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  − 2·4-s + 16-s + 2·25-s + 4·97-s − 4·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 2·4-s + 16-s + 2·25-s + 4·97-s − 4·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

Λ(s)=((316⋅1116)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((316⋅1116)s/2 ΓC(s)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 316⋅11163^{16} \cdot 11^{16}
Sign: 11
Analytic conductor: 0.007611670.00761167
Root analytic conductor: 0.7372120.737212
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 316⋅1116, ( :[0]8), 1)(16,\ 3^{16} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.45261770030.4526177003
L(12)L(\frac12) ≈\approx 0.45261770030.4526177003
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 1
11 1 1
good2 (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}
5 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
7 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
13 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
17 (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}
19 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
23 (1+T2)8 ( 1 + T^{2} )^{8}
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−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
37 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
41 (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}
43 (1+T4)4 ( 1 + T^{4} )^{4}
47 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
53 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
59 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
61 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
67 (1+T2)8 ( 1 + T^{2} )^{8}
71 (1−T2+T4−T6+T8)2 ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}
73 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
79 1−T4+T8−T12+T16 1 - T^{4} + T^{8} - T^{12} + T^{16}
83 (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}
89 (1+T2)8 ( 1 + T^{2} )^{8}
97 (1−T+T2−T3+T4)4 ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}
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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

−4.62295279559899667408415720097, −4.36092048291769080420979355836, −4.09836508095924963361008301762, −3.98516837862355863464745061497, −3.95937128375413067679711817758, −3.90734582682606064973369848354, −3.89275023384104356986805901376, −3.66785063532021729064990312177, −3.39370624573737144402028576322, −3.17705554271808749388478793727, −3.09317449994281244609463585816, −3.05361966177090638218079065431, −3.00389255426501251501802060626, −2.90227370705713524909050056627, −2.51332928671590422618556932016, −2.31917304404941302629406842304, −2.26437847382682777689274174188, −2.14535273795654806424150270438, −1.94848483513618685276119927009, −1.61142701254934689006492194806, −1.56006069287549804082634985708, −1.44427171151413130515009200183, −0.876610300552967539927896933030, −0.76198671179119282246770457432, −0.73652379186982093346515039048, 0.73652379186982093346515039048, 0.76198671179119282246770457432, 0.876610300552967539927896933030, 1.44427171151413130515009200183, 1.56006069287549804082634985708, 1.61142701254934689006492194806, 1.94848483513618685276119927009, 2.14535273795654806424150270438, 2.26437847382682777689274174188, 2.31917304404941302629406842304, 2.51332928671590422618556932016, 2.90227370705713524909050056627, 3.00389255426501251501802060626, 3.05361966177090638218079065431, 3.09317449994281244609463585816, 3.17705554271808749388478793727, 3.39370624573737144402028576322, 3.66785063532021729064990312177, 3.89275023384104356986805901376, 3.90734582682606064973369848354, 3.95937128375413067679711817758, 3.98516837862355863464745061497, 4.09836508095924963361008301762, 4.36092048291769080420979355836, 4.62295279559899667408415720097

Graph of the ZZ-function along the critical line

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