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Properties

Label 12-191e6-1.1-c0e6-0-0
Degree $12$
Conductor $4.855\times 10^{13}$
Sign $1$
Analytic cond. $7.50141\times 10^{-7}$
Root an. cond. $0.308741$
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  − 2-s − 3-s − 5-s + 6-s + 10-s − 13-s + 15-s − 17-s − 23-s + 26-s − 30-s + 34-s + 39-s − 43-s + 46-s + 6·49-s + 51-s − 59-s + 65-s − 67-s + 69-s − 78-s − 79-s + 85-s + 86-s − 97-s − 6·98-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s − 13-s + 15-s − 17-s − 23-s + 26-s − 30-s + 34-s + 39-s − 43-s + 46-s + 6·49-s + 51-s − 59-s + 65-s − 67-s + 69-s − 78-s − 79-s + 85-s + 86-s − 97-s − 6·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(191^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(191^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(191^{6}\)
Sign: $1$
Analytic conductor: \(7.50141\times 10^{-7}\)
Root analytic conductor: \(0.308741\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 191^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03382174090\)
\(L(\frac12)\) \(\approx\) \(0.03382174090\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( ( 1 - T )^{6} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
5 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
7 \( ( 1 - T )^{6}( 1 + T )^{6} \)
11 \( ( 1 - T )^{6}( 1 + T )^{6} \)
13 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
17 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
31 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
47 \( ( 1 - T )^{6}( 1 + T )^{6} \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
61 \( ( 1 - T )^{6}( 1 + T )^{6} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( ( 1 - T )^{6}( 1 + T )^{6} \)
97 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
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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

−7.35796558228346119993956786314, −7.07514261944913462684112119008, −6.93276016775063551089341074487, −6.81447333985775240623155062904, −6.48428091039454131637744410603, −6.20695986592868467970909154318, −6.11032935774456552974590701160, −5.83685122472610261452929904534, −5.65147090027015013068901858174, −5.54310332007677796858925474289, −5.28434755340001075565625085402, −5.13598570145672526331024892854, −4.70242859600969363619897054347, −4.62975578232809777521818469655, −4.24711670414528919623661361277, −4.07989511684934844691238254281, −3.97532789972335224914075549290, −3.93413846072145414738797452032, −3.19624400358957913669525606657, −3.14269715909288612675198786696, −2.80586435637208252376320780643, −2.44268081204216981902401056838, −2.00655896564504903218887133025, −1.98100675197109451299394880099, −1.06036846736001578998381077279, 1.06036846736001578998381077279, 1.98100675197109451299394880099, 2.00655896564504903218887133025, 2.44268081204216981902401056838, 2.80586435637208252376320780643, 3.14269715909288612675198786696, 3.19624400358957913669525606657, 3.93413846072145414738797452032, 3.97532789972335224914075549290, 4.07989511684934844691238254281, 4.24711670414528919623661361277, 4.62975578232809777521818469655, 4.70242859600969363619897054347, 5.13598570145672526331024892854, 5.28434755340001075565625085402, 5.54310332007677796858925474289, 5.65147090027015013068901858174, 5.83685122472610261452929904534, 6.11032935774456552974590701160, 6.20695986592868467970909154318, 6.48428091039454131637744410603, 6.81447333985775240623155062904, 6.93276016775063551089341074487, 7.07514261944913462684112119008, 7.35796558228346119993956786314

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

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