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Properties

Label 18-2252e9-563.562-c0e9-0-0
Degree $18$
Conductor $1.490\times 10^{30}$
Sign $1$
Analytic cond. $2.86107$
Root an. cond. $1.06013$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 563^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 563^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2^{18} \cdot 563^{9}\)
Sign: $1$
Analytic conductor: \(2.86107\)
Root analytic conductor: \(1.06013\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2252} (1125, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2^{18} \cdot 563^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.580605799\)
\(L(\frac12)\) \(\approx\) \(2.580605799\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
563 \( ( 1 - T )^{9} \)
good3 \( 1 + T^{9} + T^{18} \)
5 \( ( 1 - T )^{9}( 1 + T )^{9} \)
7 \( 1 + T^{9} + T^{18} \)
11 \( ( 1 + T^{3} + T^{6} )^{3} \)
13 \( ( 1 + T^{3} + T^{6} )^{3} \)
17 \( 1 + T^{9} + T^{18} \)
19 \( 1 + T^{9} + T^{18} \)
23 \( 1 + T^{9} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 - T )^{9}( 1 + T )^{9} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( ( 1 - T )^{9}( 1 + T )^{9} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( 1 + T^{9} + T^{18} \)
53 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( 1 + T^{9} + T^{18} \)
61 \( 1 + T^{9} + T^{18} \)
67 \( 1 + T^{9} + T^{18} \)
71 \( 1 + T^{9} + T^{18} \)
73 \( ( 1 - T )^{9}( 1 + T )^{9} \)
79 \( ( 1 - T )^{9}( 1 + T )^{9} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
89 \( ( 1 - T )^{9}( 1 + T )^{9} \)
97 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line

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