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Properties

Label 36-1308e18-1.1-c0e18-0-0
Degree $36$
Conductor $1.256\times 10^{56}$
Sign $1$
Analytic cond. $0.000463199$
Root an. cond. $0.807946$
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  − 9·37-s − 9·103-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 + ⋯
L(s)  = 1  − 9·37-s − 9·103-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{18} \cdot 109^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{18} \cdot 109^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(36\)
Conductor: \(2^{36} \cdot 3^{18} \cdot 109^{18}\)
Sign: $1$
Analytic conductor: \(0.000463199\)
Root analytic conductor: \(0.807946\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((36,\ 2^{36} \cdot 3^{18} \cdot 109^{18} ,\ ( \ : [0]^{18} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005581973664\)
\(L(\frac12)\) \(\approx\) \(0.005581973664\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{9} + T^{18} \)
109 \( ( 1 + T^{3} + T^{6} )^{3} \)
good5 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
7 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
11 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
13 \( ( 1 + T^{9} + T^{18} )^{2} \)
17 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
19 \( ( 1 + T^{9} + T^{18} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
29 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
31 \( ( 1 + T^{9} + T^{18} )^{2} \)
37 \( ( 1 + T + T^{2} )^{9}( 1 + T^{9} + T^{18} ) \)
41 \( ( 1 - T + T^{2} )^{9}( 1 + T + T^{2} )^{9} \)
43 \( ( 1 + T^{9} + T^{18} )^{2} \)
47 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
53 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
59 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
67 \( ( 1 + T^{9} + T^{18} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
73 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
79 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
83 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
89 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
97 \( ( 1 + T^{9} + T^{18} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.51342943323539159441584973385, −2.48721283092995269084658738544, −2.42275712637185495850346255579, −2.35663646201259193751841974185, −2.29670203546426785353423514829, −2.26561269549831114581718344788, −2.11250198831587317995889418086, −2.08233383945904163607239888144, −1.85869137579503141883335364562, −1.84309293187885968643649920898, −1.78324400854220865334922989342, −1.77956701873803275154218257481, −1.74595322695743961579296540574, −1.62915004074114295589855704760, −1.50605253044803074847824148325, −1.42323454825668234552289517878, −1.36113995090307871179337014626, −1.29742085697359210839630842039, −1.19426086453071379220434390573, −1.06485833244991883627514993836, −1.04389036731677420392857146296, −1.02974680121961494125434457040, −1.02894993229246689144170618559, −0.37591256739379681702590346682, −0.04663742405463865531376959136, 0.04663742405463865531376959136, 0.37591256739379681702590346682, 1.02894993229246689144170618559, 1.02974680121961494125434457040, 1.04389036731677420392857146296, 1.06485833244991883627514993836, 1.19426086453071379220434390573, 1.29742085697359210839630842039, 1.36113995090307871179337014626, 1.42323454825668234552289517878, 1.50605253044803074847824148325, 1.62915004074114295589855704760, 1.74595322695743961579296540574, 1.77956701873803275154218257481, 1.78324400854220865334922989342, 1.84309293187885968643649920898, 1.85869137579503141883335364562, 2.08233383945904163607239888144, 2.11250198831587317995889418086, 2.26561269549831114581718344788, 2.29670203546426785353423514829, 2.35663646201259193751841974185, 2.42275712637185495850346255579, 2.48721283092995269084658738544, 2.51342943323539159441584973385

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

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