↑

Properties

Label 18-1231e9-1231.1230-c0e9-0-0
Degree $18$
Conductor $6.491\times 10^{27}$
Sign $1$
Analytic cond. $0.0124662$
Root an. cond. $0.783804$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 9·9-s − 9·11-s + 45·81-s − 81·99-s + 36·121-s − 3·125-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 + ⋯
L(s)  = 1  + 9·9-s − 9·11-s + 45·81-s − 81·99-s + 36·121-s − 3·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1231^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1231^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(1231^{9}\)
Sign: $1$
Analytic conductor: \(0.0124662\)
Root analytic conductor: \(0.783804\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1231} (1230, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 1231^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020204729\)
\(L(\frac12)\) \(\approx\) \(1.020204729\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1231 \( ( 1 - T )^{9} \)
good2 \( 1 + T^{9} + T^{18} \)
3 \( ( 1 - T )^{9}( 1 + T )^{9} \)
5 \( ( 1 + T^{3} + T^{6} )^{3} \)
7 \( 1 + T^{9} + T^{18} \)
11 \( ( 1 + T + T^{2} )^{9} \)
13 \( 1 + T^{9} + T^{18} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( 1 + T^{9} + T^{18} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( 1 + T^{9} + T^{18} \)
31 \( 1 + T^{9} + T^{18} \)
37 \( 1 + T^{9} + T^{18} \)
41 \( ( 1 + T^{3} + T^{6} )^{3} \)
43 \( 1 + T^{9} + T^{18} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( ( 1 - T )^{9}( 1 + T )^{9} \)
61 \( ( 1 - T )^{9}( 1 + T )^{9} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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 \( ( 1 - T )^{9}( 1 + T )^{9} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.07050807193790152805794230708, −3.98953800957068807004764085591, −3.94129093183214650654431639324, −3.79891695791888238425725397052, −3.65842163410070198444329343305, −3.54392702770010665107888378622, −3.35007976591417720130544691342, −3.09676090257665378141499944733, −3.08653982574192236552040030317, −2.91842069071775297032371020689, −2.83381129697977903833987383107, −2.61479930184212261364147563914, −2.51590730218013224964821478321, −2.44592759454280781024526174074, −2.26419953767933857565612073433, −2.24304184492118921224254544458, −1.97270572324976271889704481617, −1.85626242171254914029837963548, −1.81485133252442433225241432872, −1.74299005132515802554900071663, −1.38997760743280179312350544517, −1.22589870290827693519469929606, −1.13861655597210217583624361786, −0.943898240394586313257273430629, −0.55418433231011362117618300890, 0.55418433231011362117618300890, 0.943898240394586313257273430629, 1.13861655597210217583624361786, 1.22589870290827693519469929606, 1.38997760743280179312350544517, 1.74299005132515802554900071663, 1.81485133252442433225241432872, 1.85626242171254914029837963548, 1.97270572324976271889704481617, 2.24304184492118921224254544458, 2.26419953767933857565612073433, 2.44592759454280781024526174074, 2.51590730218013224964821478321, 2.61479930184212261364147563914, 2.83381129697977903833987383107, 2.91842069071775297032371020689, 3.08653982574192236552040030317, 3.09676090257665378141499944733, 3.35007976591417720130544691342, 3.54392702770010665107888378622, 3.65842163410070198444329343305, 3.79891695791888238425725397052, 3.94129093183214650654431639324, 3.98953800957068807004764085591, 4.07050807193790152805794230708

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

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