Properties

Label 18-1999e9-1999.1998-c0e9-0-0
Degree $18$
Conductor $5.097\times 10^{29}$
Sign $1$
Analytic cond. $0.978879$
Root an. cond. $0.998814$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s − 9·79-s + 45·81-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 + ⋯
L(s)  = 1  − 3·8-s + 9·9-s + 9·49-s + 3·64-s − 27·72-s − 9·79-s + 45·81-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1999^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1999^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(1999^{9}\)
Sign: $1$
Analytic conductor: \(0.978879\)
Root analytic conductor: \(0.998814\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1999} (1998, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 1999^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.017735772\)
\(L(\frac12)\) \(\approx\) \(4.017735772\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1999 \( ( 1 - T )^{9} \)
good2 \( ( 1 + T^{3} + T^{6} )^{3} \)
3 \( ( 1 - T )^{9}( 1 + T )^{9} \)
5 \( 1 + T^{9} + T^{18} \)
7 \( ( 1 - T )^{9}( 1 + T )^{9} \)
11 \( 1 + T^{9} + T^{18} \)
13 \( 1 + T^{9} + T^{18} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( 1 + T^{9} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( 1 + T^{9} + T^{18} \)
37 \( 1 + T^{9} + T^{18} \)
41 \( ( 1 + T^{3} + T^{6} )^{3} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( 1 + T^{9} + T^{18} \)
59 \( 1 + T^{9} + T^{18} \)
61 \( 1 + T^{9} + T^{18} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( 1 + T^{9} + T^{18} \)
73 \( ( 1 - T )^{9}( 1 + T )^{9} \)
79 \( ( 1 + T + T^{2} )^{9} \)
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

−3.79176762952031764320270421991, −3.75399302478977961172276974199, −3.67011388962456349578618538585, −3.62941793363368733333421194876, −3.59300015773679058159407408618, −3.20423107783851481412107319081, −2.92336681090380113213081444336, −2.90558538318245228780441084833, −2.85763883781879452377108217141, −2.80772190251028919032264138316, −2.48971949477049115945887030453, −2.44973823615711600982815263934, −2.39422333146600656434440557447, −2.30643809889651343823578983846, −2.05411147918019502541854764261, −1.89838391853538569543298357597, −1.86448843237071398693566789467, −1.63498344347997472894935154832, −1.41156201859132925349198587018, −1.35895581883200150744010758821, −1.30659825116056007744396350025, −1.25276434109426018947382419609, −0.960235859488580258087108771962, −0.799026337350778006674919180746, −0.77478281085216062984192416276, 0.77478281085216062984192416276, 0.799026337350778006674919180746, 0.960235859488580258087108771962, 1.25276434109426018947382419609, 1.30659825116056007744396350025, 1.35895581883200150744010758821, 1.41156201859132925349198587018, 1.63498344347997472894935154832, 1.86448843237071398693566789467, 1.89838391853538569543298357597, 2.05411147918019502541854764261, 2.30643809889651343823578983846, 2.39422333146600656434440557447, 2.44973823615711600982815263934, 2.48971949477049115945887030453, 2.80772190251028919032264138316, 2.85763883781879452377108217141, 2.90558538318245228780441084833, 2.92336681090380113213081444336, 3.20423107783851481412107319081, 3.59300015773679058159407408618, 3.62941793363368733333421194876, 3.67011388962456349578618538585, 3.75399302478977961172276974199, 3.79176762952031764320270421991

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

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