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Properties

Label 12-2888e6-1.1-c0e6-0-6
Degree $12$
Conductor $5.802\times 10^{20}$
Sign $1$
Analytic cond. $8.96449$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 8-s + 3·9-s + 3·24-s − 27-s + 3·41-s − 3·49-s + 3·59-s + 3·67-s + 3·72-s + 6·73-s − 6·81-s + 3·97-s − 3·107-s + 9·123-s + 127-s + 131-s + 137-s + 139-s − 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 9·177-s + ⋯
L(s)  = 1  + 3·3-s + 8-s + 3·9-s + 3·24-s − 27-s + 3·41-s − 3·49-s + 3·59-s + 3·67-s + 3·72-s + 6·73-s − 6·81-s + 3·97-s − 3·107-s + 9·123-s + 127-s + 131-s + 137-s + 139-s − 9·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 9·177-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 19^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.295290659\)
\(L(\frac12)\) \(\approx\) \(6.295290659\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good3 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
11 \( ( 1 + T^{3} + T^{6} )^{2} \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 + T^{3} + T^{6} )^{2} \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
59 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
71 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
73 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 + T^{3} + T^{6} )^{2} \)
89 \( ( 1 - T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
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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

−4.59695926190580993289383868058, −4.58292342042425636953078120841, −4.35888337750816924818587293171, −4.34002497035877594857825361573, −3.98539576257835080118937488202, −3.77082388704072490170544603374, −3.76995785057853086077639489684, −3.73699523559635279802040269465, −3.55829227790061167355420313929, −3.49837157215907245605876715326, −3.26232174611936357594228154377, −2.93725894377190229711552888042, −2.91581103553871493582893223457, −2.85664646303961472104627282287, −2.61097146011034204308030634218, −2.30655533374705216704109289811, −2.28241363928537793495242523893, −2.15111313930195913063104150678, −2.08253507639876977390540712514, −1.91505796115488108838114216587, −1.75921536093604927199288772808, −1.17052212995568422422147514703, −1.10223040610878316414576676021, −1.00744616246581016857511710843, −0.60647879965516300751839247632, 0.60647879965516300751839247632, 1.00744616246581016857511710843, 1.10223040610878316414576676021, 1.17052212995568422422147514703, 1.75921536093604927199288772808, 1.91505796115488108838114216587, 2.08253507639876977390540712514, 2.15111313930195913063104150678, 2.28241363928537793495242523893, 2.30655533374705216704109289811, 2.61097146011034204308030634218, 2.85664646303961472104627282287, 2.91581103553871493582893223457, 2.93725894377190229711552888042, 3.26232174611936357594228154377, 3.49837157215907245605876715326, 3.55829227790061167355420313929, 3.73699523559635279802040269465, 3.76995785057853086077639489684, 3.77082388704072490170544603374, 3.98539576257835080118937488202, 4.34002497035877594857825361573, 4.35888337750816924818587293171, 4.58292342042425636953078120841, 4.59695926190580993289383868058

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

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