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Properties

Label 12-1859e6-1.1-c0e6-0-2
Degree $12$
Conductor $4.127\times 10^{19}$
Sign $1$
Analytic cond. $0.637703$
Root an. cond. $0.963203$
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  + 3-s − 3·4-s + 2·5-s + 9-s + 3·11-s − 3·12-s + 2·15-s + 3·16-s − 6·20-s + 23-s + 25-s + 2·31-s + 3·33-s − 3·36-s − 37-s − 9·44-s + 2·45-s + 2·47-s + 3·48-s − 3·49-s − 2·53-s + 6·55-s − 59-s − 6·60-s + 2·64-s + 6·67-s + 69-s + ⋯
L(s)  = 1  + 3-s − 3·4-s + 2·5-s + 9-s + 3·11-s − 3·12-s + 2·15-s + 3·16-s − 6·20-s + 23-s + 25-s + 2·31-s + 3·33-s − 3·36-s − 37-s − 9·44-s + 2·45-s + 2·47-s + 3·48-s − 3·49-s − 2·53-s + 6·55-s − 59-s − 6·60-s + 2·64-s + 6·67-s + 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(0.637703\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 11^{6} \cdot 13^{12} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.687648395\)
\(L(\frac12)\) \(\approx\) \(1.687648395\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 - T + T^{2} )^{3} \)
13 \( 1 \)
good2 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
7 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
17 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
19 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
23 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
29 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
37 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
43 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
59 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
61 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
67 \( ( 1 - T + T^{2} )^{6} \)
71 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
97 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
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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

−4.91764685008333796867960319237, −4.70851902956968859427785684825, −4.58326091166527361973185677157, −4.57585311094722520186872946008, −4.54513207177709199599726641901, −4.47331552860339295384790494166, −4.01785958866756952172148102533, −3.96021448402681239518612304736, −3.79894262483043957234214970432, −3.74213256637403067702302565758, −3.58248623062528508787844967306, −3.48594262308510766057951267835, −3.27803989427556190133315264290, −3.01778991815372593514134567190, −2.66320411705426773043759822256, −2.51502387273918812684842421364, −2.49860207756640481287262127317, −2.36536180799443466949155882835, −1.90621716319244380771349774574, −1.68083895241522822737107859156, −1.65550912931061793235162238534, −1.47573279448361349308033880185, −1.20862475677991773526532489167, −0.910403748579808583283472202124, −0.63888356687725366145910625436, 0.63888356687725366145910625436, 0.910403748579808583283472202124, 1.20862475677991773526532489167, 1.47573279448361349308033880185, 1.65550912931061793235162238534, 1.68083895241522822737107859156, 1.90621716319244380771349774574, 2.36536180799443466949155882835, 2.49860207756640481287262127317, 2.51502387273918812684842421364, 2.66320411705426773043759822256, 3.01778991815372593514134567190, 3.27803989427556190133315264290, 3.48594262308510766057951267835, 3.58248623062528508787844967306, 3.74213256637403067702302565758, 3.79894262483043957234214970432, 3.96021448402681239518612304736, 4.01785958866756952172148102533, 4.47331552860339295384790494166, 4.54513207177709199599726641901, 4.57585311094722520186872946008, 4.58326091166527361973185677157, 4.70851902956968859427785684825, 4.91764685008333796867960319237

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

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