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Properties

Label 36-648e18-1.1-c0e18-0-0
Degree $36$
Conductor $4.058\times 10^{50}$
Sign $1$
Analytic cond. $1.49685\times 10^{-9}$
Root an. cond. $0.568677$
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  − 9·59-s − 9·89-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  − 9·59-s − 9·89-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(36\)
Conductor: \(2^{54} \cdot 3^{72}\)
Sign: $1$
Analytic conductor: \(1.49685\times 10^{-9}\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((36,\ 2^{54} \cdot 3^{72} ,\ ( \ : [0]^{18} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1219254973\)
\(L(\frac12)\) \(\approx\) \(0.1219254973\)
\(L(1)\) 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 \( 1 + T^{9} + T^{18} \)
good5 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
7 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
11 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
13 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
17 \( ( 1 + T^{9} + T^{18} )^{2} \)
19 \( ( 1 + T^{9} + T^{18} )^{2} \)
23 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
29 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
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 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
47 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
53 \( ( 1 - T + T^{2} )^{9}( 1 + T + T^{2} )^{9} \)
59 \( ( 1 + T + T^{2} )^{9}( 1 + T^{9} + T^{18} ) \)
61 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
67 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
71 \( ( 1 - T^{3} + T^{6} )^{3}( 1 + T^{3} + T^{6} )^{3} \)
73 \( ( 1 + T^{9} + T^{18} )^{2} \)
79 \( ( 1 - T^{9} + T^{18} )( 1 + T^{9} + T^{18} ) \)
83 \( ( 1 + T^{9} + T^{18} )^{2} \)
89 \( ( 1 + T + T^{2} )^{9}( 1 + T^{3} + T^{6} )^{3} \)
97 \( ( 1 + T^{3} + T^{6} )^{3}( 1 + T^{9} + T^{18} ) \)
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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

−2.91612837821210557776232741429, −2.86034020251550513474836739733, −2.80059202278079678370494757819, −2.61078885212782939094053098120, −2.60807273551117587801918806412, −2.57520447454442846389737597677, −2.52149860568947923448255700738, −2.51804787754036702263103337850, −2.43172516855255334981062552482, −2.41066466162424057981024329576, −2.05103618923782202105829402726, −1.94171306769174556875580755088, −1.80172039958653432120378480743, −1.76495102094618918388633879513, −1.74783078472615069124938133200, −1.67990703480025224177193494747, −1.66458176566064431164403609055, −1.63438328148246247318544547598, −1.62119570828251856248690417829, −1.38273988979150669222123159700, −1.19277327915257926647068187499, −1.18272167151859846625180239184, −1.02034719433780609902134972522, −0.936710690892133042356387586057, −0.62189711512920072736902268774, 0.62189711512920072736902268774, 0.936710690892133042356387586057, 1.02034719433780609902134972522, 1.18272167151859846625180239184, 1.19277327915257926647068187499, 1.38273988979150669222123159700, 1.62119570828251856248690417829, 1.63438328148246247318544547598, 1.66458176566064431164403609055, 1.67990703480025224177193494747, 1.74783078472615069124938133200, 1.76495102094618918388633879513, 1.80172039958653432120378480743, 1.94171306769174556875580755088, 2.05103618923782202105829402726, 2.41066466162424057981024329576, 2.43172516855255334981062552482, 2.51804787754036702263103337850, 2.52149860568947923448255700738, 2.57520447454442846389737597677, 2.60807273551117587801918806412, 2.61078885212782939094053098120, 2.80059202278079678370494757819, 2.86034020251550513474836739733, 2.91612837821210557776232741429

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

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