Normalization:  

Dirichlet series

L(s)  = 1  − 6·19-s + 3·37-s + 3·73-s − 6·109-s + ⋯
L(s)  = 1  − 6·19-s + 3·37-s + 3·73-s − 6·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{36}\)
Sign: $1$
Analytic conductor: \(9.49878\)
Root analytic conductor: \(1.20634\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{36} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03879708614\)
\(L(\frac12)\) \(\approx\) \(0.03879708614\)
\(L(1)\) 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 \( 1 \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 + T^{3} + T^{6} )^{2} \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 + T + T^{2} )^{6} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 + T^{3} + T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.71851718919101419715562853271, −4.55590576807730731994737253149, −4.36677181808462205386167842181, −4.24677184440665543676023549659, −4.22317279401170678449285038824, −4.18173550124059446559257178701, −3.92762110810703715687938416074, −3.70202208893669596103443204989, −3.63763504124611025364846249104, −3.41243880417043443988725364894, −3.35521092686692162696325543355, −3.08797613226390450602248325129, −2.85538139248227052526624223090, −2.59508240641849098155244636314, −2.49360157203146396777230436039, −2.32072647263946060697173543796, −2.23554933301303663513992845668, −2.17059485469055116819994071005, −2.16338642521178496328017446251, −1.72469264365200716840682560887, −1.47517846044465817875606327579, −1.29237330693346055425959769201, −0.936576595728922874905163411600, −0.931405183104292041692758036452, −0.06471887733229646039410735002, 0.06471887733229646039410735002, 0.931405183104292041692758036452, 0.936576595728922874905163411600, 1.29237330693346055425959769201, 1.47517846044465817875606327579, 1.72469264365200716840682560887, 2.16338642521178496328017446251, 2.17059485469055116819994071005, 2.23554933301303663513992845668, 2.32072647263946060697173543796, 2.49360157203146396777230436039, 2.59508240641849098155244636314, 2.85538139248227052526624223090, 3.08797613226390450602248325129, 3.35521092686692162696325543355, 3.41243880417043443988725364894, 3.63763504124611025364846249104, 3.70202208893669596103443204989, 3.92762110810703715687938416074, 4.18173550124059446559257178701, 4.22317279401170678449285038824, 4.24677184440665543676023549659, 4.36677181808462205386167842181, 4.55590576807730731994737253149, 4.71851718919101419715562853271

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

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