Properties

Label 12-3e36-1.1-c1e6-0-13
Degree $12$
Conductor $1.501\times 10^{17}$
Sign $1$
Analytic cond. $38907.0$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s − 12·7-s − 12·13-s + 15·16-s − 24·19-s − 15·25-s + 72·28-s − 30·31-s − 6·37-s − 12·43-s + 48·49-s + 72·52-s − 12·61-s − 13·64-s + 6·67-s + 12·73-s + 144·76-s − 48·79-s + 144·91-s − 12·97-s + 90·100-s − 12·103-s + 12·109-s − 180·112-s − 24·121-s + 180·124-s + 127-s + ⋯
L(s)  = 1  − 3·4-s − 4.53·7-s − 3.32·13-s + 15/4·16-s − 5.50·19-s − 3·25-s + 13.6·28-s − 5.38·31-s − 0.986·37-s − 1.82·43-s + 48/7·49-s + 9.98·52-s − 1.53·61-s − 1.62·64-s + 0.733·67-s + 1.40·73-s + 16.5·76-s − 5.40·79-s + 15.0·91-s − 1.21·97-s + 9·100-s − 1.18·103-s + 1.14·109-s − 17.0·112-s − 2.18·121-s + 16.1·124-s + 0.0887·127-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{36}\)
Sign: $1$
Analytic conductor: \(38907.0\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 3 p T^{2} + 21 T^{4} + 49 T^{6} + 21 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \)
5 \( 1 + 3 p T^{2} + 93 T^{4} + 427 T^{6} + 93 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12} \)
7 \( ( 1 + 6 T + 30 T^{2} + 85 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( 1 + 24 T^{2} + 372 T^{4} + 3955 T^{6} + 372 p^{2} T^{8} + 24 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 6 T + 30 T^{2} + 85 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 21 T^{2} + 582 T^{4} + 7729 T^{6} + 582 p^{2} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 12 T + 96 T^{2} + 25 p T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 24 T^{2} + 696 T^{4} + 30091 T^{6} + 696 p^{2} T^{8} + 24 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 123 T^{2} + 7509 T^{4} + 273307 T^{6} + 7509 p^{2} T^{8} + 123 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 + 15 T + 156 T^{2} + 1003 T^{3} + 156 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 3 T + 105 T^{2} + 205 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 159 T^{2} + 12801 T^{4} + 644539 T^{6} + 12801 p^{2} T^{8} + 159 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 + 6 T + 3 p T^{2} + 508 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 267 T^{2} + 30369 T^{4} + 1882723 T^{6} + 30369 p^{2} T^{8} + 267 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 + 210 T^{2} + 21831 T^{4} + 1416508 T^{6} + 21831 p^{2} T^{8} + 210 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 168 T^{2} + 9876 T^{4} + 406507 T^{6} + 9876 p^{2} T^{8} + 168 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 6 T + 111 T^{2} + 436 T^{3} + 111 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 3 T + 57 T^{2} - 653 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( 1 + 354 T^{2} + 56463 T^{4} + 5162812 T^{6} + 56463 p^{2} T^{8} + 354 p^{4} T^{10} + p^{6} T^{12} \)
73 \( ( 1 - 6 T + 150 T^{2} - 965 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 + 24 T + 372 T^{2} + 3685 T^{3} + 372 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 60 T^{2} + 9696 T^{4} + 375547 T^{6} + 9696 p^{2} T^{8} + 60 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 147 T^{2} - 2811 T^{4} - 1426997 T^{6} - 2811 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12} \)
97 \( ( 1 + 6 T + 192 T^{2} + 517 T^{3} + 192 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.02778339142187332055895830152, −5.75524586441328100705071692080, −5.75522192822413679222060617235, −5.42491078164367133477291044600, −5.39442895080689314598392697047, −5.27003126035738856807944713383, −5.17535371438318617054043815800, −4.61866102789252945367366014863, −4.60382592339520975320540338521, −4.39161919178226468263374453713, −4.38400877958290081853309367109, −4.23515360459893042182284278796, −4.11804420121259399366825220900, −3.67069318112554478018041202332, −3.52835046676554308811434274525, −3.48319650034324299815555660693, −3.46729562625808468602228415298, −3.37271331776158128599555717434, −2.68660868794373027615249931288, −2.62645260948605427964741843165, −2.47101529622553094338710593714, −2.25062693712165773401531433440, −1.91423116775228000069691215094, −1.72010988515115534229040122010, −1.60267574387908682091371094118, 0, 0, 0, 0, 0, 0, 1.60267574387908682091371094118, 1.72010988515115534229040122010, 1.91423116775228000069691215094, 2.25062693712165773401531433440, 2.47101529622553094338710593714, 2.62645260948605427964741843165, 2.68660868794373027615249931288, 3.37271331776158128599555717434, 3.46729562625808468602228415298, 3.48319650034324299815555660693, 3.52835046676554308811434274525, 3.67069318112554478018041202332, 4.11804420121259399366825220900, 4.23515360459893042182284278796, 4.38400877958290081853309367109, 4.39161919178226468263374453713, 4.60382592339520975320540338521, 4.61866102789252945367366014863, 5.17535371438318617054043815800, 5.27003126035738856807944713383, 5.39442895080689314598392697047, 5.42491078164367133477291044600, 5.75522192822413679222060617235, 5.75524586441328100705071692080, 6.02778339142187332055895830152

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

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