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Properties

Label 18-2099e9-2099.2098-c0e9-0-0
Degree $18$
Conductor $7.909\times 10^{29}$
Sign $1$
Analytic cond. $1.51888$
Root an. cond. $1.02349$
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-s + 9·4-s − 5-s − 7-s − 9·12-s + 15-s + 45·16-s − 17-s − 9·20-s + 21-s − 9·28-s − 31-s + 35-s − 37-s − 41-s − 47-s − 45·48-s + 51-s − 59-s + 9·60-s − 61-s + 165·64-s − 9·68-s − 73-s − 45·80-s − 83-s + 9·84-s + ⋯
L(s)  = 1  − 3-s + 9·4-s − 5-s − 7-s − 9·12-s + 15-s + 45·16-s − 17-s − 9·20-s + 21-s − 9·28-s − 31-s + 35-s − 37-s − 41-s − 47-s − 45·48-s + 51-s − 59-s + 9·60-s − 61-s + 165·64-s − 9·68-s − 73-s − 45·80-s − 83-s + 9·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2099^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2099^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(2099^{9}\)
Sign: $1$
Analytic conductor: \(1.51888\)
Root analytic conductor: \(1.02349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2099} (2098, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 2099^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(11.88462096\)
\(L(\frac12)\) \(\approx\) \(11.88462096\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2099 \( 1+O(T) \)
good2 \( ( 1 - T )^{9}( 1 + T )^{9} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
5 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
37 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
41 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
53 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
61 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
79 \( ( 1 - T )^{9}( 1 + T )^{9} \)
83 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
89 \( ( 1 - T )^{9}( 1 + T )^{9} \)
97 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
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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

−3.53920319872386738154572517214, −3.37400808102350461423826435961, −3.33991379077803202220847743822, −3.28739581704687977272578357581, −3.28536218071652096840082786552, −3.19136628969817546969524955923, −3.00237780012377806065042923640, −2.84987458017727338140790084137, −2.74162744010211009644746761532, −2.69130598126520852091392945743, −2.55346078014841485868578168382, −2.47137678254500778257554924958, −2.45325998617763253536984806056, −2.16500553495065459304966047708, −2.06091012209795234722583541437, −1.96415178961484087092421439858, −1.83839679074803914683264147311, −1.79202523083265659109235408277, −1.77158910709321354217176228645, −1.59519908111296362059832053118, −1.24657016988247887600040525354, −1.22690456157783011660046725026, −1.20148953792791440111627602405, −1.06076781320537743708094615969, −0.60007384485686042510774962728, 0.60007384485686042510774962728, 1.06076781320537743708094615969, 1.20148953792791440111627602405, 1.22690456157783011660046725026, 1.24657016988247887600040525354, 1.59519908111296362059832053118, 1.77158910709321354217176228645, 1.79202523083265659109235408277, 1.83839679074803914683264147311, 1.96415178961484087092421439858, 2.06091012209795234722583541437, 2.16500553495065459304966047708, 2.45325998617763253536984806056, 2.47137678254500778257554924958, 2.55346078014841485868578168382, 2.69130598126520852091392945743, 2.74162744010211009644746761532, 2.84987458017727338140790084137, 3.00237780012377806065042923640, 3.19136628969817546969524955923, 3.28536218071652096840082786552, 3.28739581704687977272578357581, 3.33991379077803202220847743822, 3.37400808102350461423826435961, 3.53920319872386738154572517214

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

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