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Properties

Label 36-1620e18-1.1-c0e18-0-1
Degree $36$
Conductor $5.906\times 10^{57}$
Sign $1$
Analytic cond. $0.0217820$
Root an. cond. $0.899158$
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  + 18·23-s − 9·41-s + 18·67-s − 9·83-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 + ⋯
L(s)  = 1  + 18·23-s − 9·41-s + 18·67-s − 9·83-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{72} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{72} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(36\)
Conductor: \(2^{36} \cdot 3^{72} \cdot 5^{18}\)
Sign: $1$
Analytic conductor: \(0.0217820\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((36,\ 2^{36} \cdot 3^{72} \cdot 5^{18} ,\ ( \ : [0]^{18} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.653537361\)
\(L(\frac12)\) \(\approx\) \(3.653537361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{9} + T^{18} \)
3 \( 1 + T^{9} + T^{18} \)
5 \( 1 + T^{9} + T^{18} \)
good7 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
11 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
13 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
19 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
23 \( ( 1 - T )^{18}( 1 + T^{9} + T^{18} ) \)
29 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
31 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
37 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
41 \( ( 1 + T + T^{2} )^{9}( 1 + T^{9} + T^{18} ) \)
43 \( ( 1 + T^{9} + T^{18} )^{2} \)
47 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
53 \( ( 1 - T + T^{2} )^{9}( 1 + T + T^{2} )^{9} \)
59 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
67 \( ( 1 - T )^{18}( 1 + T^{9} + T^{18} ) \)
71 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
73 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
79 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
83 \( ( 1 + T + T^{2} )^{9}( 1 + T^{9} + T^{18} ) \)
89 \( ( 1 + T^{9} + T^{18} )^{2} \)
97 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
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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

−2.34415409650363142519678841769, −2.30447945362343786712977226103, −2.30017651302046543405861657908, −2.26343276940070835628274811903, −2.25475779036283924844716479777, −2.19602695712033201719992927177, −2.05734214625405804406309294466, −2.03642538658427038926008696549, −1.92931860385832624042396779901, −1.84117353542762479266072955373, −1.60676390013566696565705723484, −1.59317685091627152973999647642, −1.52888332527984293629154593507, −1.51176194053916016582421171641, −1.40288327259528342275475808328, −1.33629984357300196680210408150, −1.22659192047751940289485707590, −1.03229455790525522683899251337, −1.02135160696691827968826201653, −0.991462626967129498985460659260, −0.976965450279633691849217498570, −0.926922958870107865487149797872, −0.829294973263739909973576254025, −0.806980295074838910009534645652, −0.65052215094857398945467890743, 0.65052215094857398945467890743, 0.806980295074838910009534645652, 0.829294973263739909973576254025, 0.926922958870107865487149797872, 0.976965450279633691849217498570, 0.991462626967129498985460659260, 1.02135160696691827968826201653, 1.03229455790525522683899251337, 1.22659192047751940289485707590, 1.33629984357300196680210408150, 1.40288327259528342275475808328, 1.51176194053916016582421171641, 1.52888332527984293629154593507, 1.59317685091627152973999647642, 1.60676390013566696565705723484, 1.84117353542762479266072955373, 1.92931860385832624042396779901, 2.03642538658427038926008696549, 2.05734214625405804406309294466, 2.19602695712033201719992927177, 2.25475779036283924844716479777, 2.26343276940070835628274811903, 2.30017651302046543405861657908, 2.30447945362343786712977226103, 2.34415409650363142519678841769

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

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