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Properties

Label 18-2252e9-563.562-c0e9-0-0
Degree 1818
Conductor 1.490×10301.490\times 10^{30}
Sign 11
Analytic cond. 2.861072.86107
Root an. cond. 1.060131.06013
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·25-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 + 239-s + 241-s + 251-s + 257-s + 263-s + ⋯
L(s)  = 1  + 9·25-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 + 239-s + 241-s + 251-s + 257-s + 263-s + ⋯

Functional equation

Λ(s)=((218⋅5639)s/2 ΓC(s)9 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 563^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((218⋅5639)s/2 ΓC(s)9 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 563^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1818
Conductor: 218⋅56392^{18} \cdot 563^{9}
Sign: 11
Analytic conductor: 2.861072.86107
Root analytic conductor: 1.060131.06013
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ2252(1125,⋅)\chi_{2252} (1125, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (18, 218⋅5639, ( :[0]9), 1)(18,\ 2^{18} \cdot 563^{9} ,\ ( \ : [0]^{9} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5806057992.580605799
L(12)L(\frac12) ≈\approx 2.5806057992.580605799
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)
bad2 1 1
563 (1−T)9 ( 1 - T )^{9}
good3 1+T9+T18 1 + T^{9} + T^{18}
5 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
7 1+T9+T18 1 + T^{9} + T^{18}
11 (1+T3+T6)3 ( 1 + T^{3} + T^{6} )^{3}
13 (1+T3+T6)3 ( 1 + T^{3} + T^{6} )^{3}
17 1+T9+T18 1 + T^{9} + T^{18}
19 1+T9+T18 1 + T^{9} + T^{18}
23 1+T9+T18 1 + T^{9} + T^{18}
29 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
31 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
37 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
41 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
43 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
47 1+T9+T18 1 + T^{9} + T^{18}
53 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
59 1+T9+T18 1 + T^{9} + T^{18}
61 1+T9+T18 1 + T^{9} + T^{18}
67 1+T9+T18 1 + T^{9} + T^{18}
71 1+T9+T18 1 + T^{9} + T^{18}
73 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
79 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
83 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
89 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
97 (1−T)9(1+T)9 ( 1 - T )^{9}( 1 + T )^{9}
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   L(s)=∏p ∏j=118(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.47539300418552690642811392844, −3.45702381767122316480661746932, −3.31454544763523863571787738310, −3.29892507157274216071246145168, −3.29776619003459381464757921300, −3.29374942974297752209877609742, −3.15147785968101158765817534433, −2.85187960741179600182163719935, −2.70162039438089637305471319333, −2.65332490613054074703632286503, −2.56672593193524420916362827496, −2.54120884557246844270772852873, −2.32588419477926953933161235694, −2.32429506822983432508869480497, −2.29261725614863923775033795529, −1.88546027104268711336205415823, −1.82937618635833034914112576186, −1.56197790536077946741891271994, −1.31905885988994266383484269465, −1.29470870699975285602584800636, −1.28416884147980740679574925073, −1.08281695375897246588041831662, −1.00228406876684600882143098439, −0.70943252120915702779488196197, −0.59101860870745974210120487340, 0.59101860870745974210120487340, 0.70943252120915702779488196197, 1.00228406876684600882143098439, 1.08281695375897246588041831662, 1.28416884147980740679574925073, 1.29470870699975285602584800636, 1.31905885988994266383484269465, 1.56197790536077946741891271994, 1.82937618635833034914112576186, 1.88546027104268711336205415823, 2.29261725614863923775033795529, 2.32429506822983432508869480497, 2.32588419477926953933161235694, 2.54120884557246844270772852873, 2.56672593193524420916362827496, 2.65332490613054074703632286503, 2.70162039438089637305471319333, 2.85187960741179600182163719935, 3.15147785968101158765817534433, 3.29374942974297752209877609742, 3.29776619003459381464757921300, 3.29892507157274216071246145168, 3.31454544763523863571787738310, 3.45702381767122316480661746932, 3.47539300418552690642811392844

Graph of the ZZ-function along the critical line

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