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Properties

Label 12-3645e6-1.1-c0e6-0-5
Degree $12$
Conductor $2.345\times 10^{21}$
Sign $1$
Analytic cond. $36.2349$
Root an. cond. $1.34873$
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  + 2·8-s − 3·17-s + 3·19-s + 6·53-s + 64-s − 12·107-s − 6·109-s + 125-s + 127-s + 131-s − 6·136-s + 137-s + 139-s + 149-s + 151-s + 6·152-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2·8-s − 3·17-s + 3·19-s + 6·53-s + 64-s − 12·107-s − 6·109-s + 125-s + 127-s + 131-s − 6·136-s + 137-s + 139-s + 149-s + 151-s + 6·152-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{36} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(36.2349\)
Root analytic conductor: \(1.34873\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{36} \cdot 5^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.029683501\)
\(L(\frac12)\) \(\approx\) \(2.029683501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T^{3} + T^{6} \)
good2 \( ( 1 - T^{3} + T^{6} )^{2} \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
19 \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \)
23 \( ( 1 - T^{3} + T^{6} )^{2} \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 + T^{3} + T^{6} )^{2} \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
43 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )^{2} \)
53 \( ( 1 - T + T^{2} )^{6} \)
59 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
61 \( ( 1 + T^{3} + T^{6} )^{2} \)
67 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
79 \( ( 1 + T^{3} + T^{6} )^{2} \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
97 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
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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

−4.44087938284334154727833098831, −4.34898727848242622195522475595, −4.31750946943726764677477736109, −4.17674372787268232015863668044, −4.01776355698430683460293546805, −3.99039677008518975206402101424, −3.90032189161582702662827503091, −3.77420088841146591466932320896, −3.33987648672607902671165217875, −3.33892953620750641831089407186, −3.20001487732255687435525563163, −2.90350869716948366315388397841, −2.77139293116896321575422762839, −2.50627194864857714700282749014, −2.43463959472793624754975509714, −2.33101783473370229030080001275, −2.30777041134171095976415363702, −2.24060517984886462858065627985, −1.51221687213511594313654647007, −1.50762449235076007186700718727, −1.40083980596807246969160995650, −1.34595772554037149266735576167, −1.21456077561508205553633504705, −0.77014015216994414075940004411, −0.40325269430920316962504812392, 0.40325269430920316962504812392, 0.77014015216994414075940004411, 1.21456077561508205553633504705, 1.34595772554037149266735576167, 1.40083980596807246969160995650, 1.50762449235076007186700718727, 1.51221687213511594313654647007, 2.24060517984886462858065627985, 2.30777041134171095976415363702, 2.33101783473370229030080001275, 2.43463959472793624754975509714, 2.50627194864857714700282749014, 2.77139293116896321575422762839, 2.90350869716948366315388397841, 3.20001487732255687435525563163, 3.33892953620750641831089407186, 3.33987648672607902671165217875, 3.77420088841146591466932320896, 3.90032189161582702662827503091, 3.99039677008518975206402101424, 4.01776355698430683460293546805, 4.17674372787268232015863668044, 4.31750946943726764677477736109, 4.34898727848242622195522475595, 4.44087938284334154727833098831

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

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