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Properties

Label 18-983e9-983.982-c0e9-0-0
Degree $18$
Conductor $8.570\times 10^{26}$
Sign $1$
Analytic cond. $0.00164587$
Root an. cond. $0.700414$
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  + 9·25-s − 9·59-s − 9·79-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-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 + ⋯
L(s)  = 1  + 9·25-s − 9·59-s − 9·79-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(983^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(983^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(983^{9}\)
Sign: $1$
Analytic conductor: \(0.00164587\)
Root analytic conductor: \(0.700414\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{983} (982, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 983^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8541104425\)
\(L(\frac12)\) \(\approx\) \(0.8541104425\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad983 \( ( 1 - T )^{9} \)
good2 \( 1 + T^{9} + T^{18} \)
3 \( 1 + T^{9} + T^{18} \)
5 \( ( 1 - T )^{9}( 1 + T )^{9} \)
7 \( 1 + T^{9} + T^{18} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( 1 + T^{9} + T^{18} \)
23 \( 1 + T^{9} + T^{18} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( ( 1 + T^{3} + T^{6} )^{3} \)
37 \( ( 1 + T^{3} + T^{6} )^{3} \)
41 \( ( 1 + T^{3} + T^{6} )^{3} \)
43 \( 1 + T^{9} + T^{18} \)
47 \( 1 + T^{9} + T^{18} \)
53 \( 1 + T^{9} + T^{18} \)
59 \( ( 1 + T + T^{2} )^{9} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( ( 1 + T^{3} + T^{6} )^{3} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( ( 1 - T )^{9}( 1 + T )^{9} \)
79 \( ( 1 + T + T^{2} )^{9} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
89 \( 1 + T^{9} + T^{18} \)
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

−4.32495861042507910678864118183, −4.27927588831186845009806735243, −4.00327508945780570494326940933, −3.84405139581806485856280208519, −3.70680546949874528582277512972, −3.36672971197555799117340141244, −3.23779736752656169076878360319, −3.23265796650335331857512417721, −3.12154282368462952980411590265, −3.10617379629877749375058361476, −3.00003615544217930872589713744, −2.95587660680363076836170497980, −2.95298206511947889534990129428, −2.60375624198040442611285012617, −2.52750703155840666162534528489, −2.43791565196890625889155297201, −2.12163706494788260628324484294, −1.92939925858628062822630998349, −1.75262783880172750470859353255, −1.53106581503334650470559092247, −1.42000795716677552711642406126, −1.37418132463718640727369348763, −1.19276263282528018267333375633, −0.914489418029993841565598819664, −0.78214809640805405585048506681, 0.78214809640805405585048506681, 0.914489418029993841565598819664, 1.19276263282528018267333375633, 1.37418132463718640727369348763, 1.42000795716677552711642406126, 1.53106581503334650470559092247, 1.75262783880172750470859353255, 1.92939925858628062822630998349, 2.12163706494788260628324484294, 2.43791565196890625889155297201, 2.52750703155840666162534528489, 2.60375624198040442611285012617, 2.95298206511947889534990129428, 2.95587660680363076836170497980, 3.00003615544217930872589713744, 3.10617379629877749375058361476, 3.12154282368462952980411590265, 3.23265796650335331857512417721, 3.23779736752656169076878360319, 3.36672971197555799117340141244, 3.70680546949874528582277512972, 3.84405139581806485856280208519, 4.00327508945780570494326940933, 4.27927588831186845009806735243, 4.32495861042507910678864118183

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

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