Properties

Label 12-114e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.195\times 10^{12}$
Sign $1$
Analytic cond. $0.568973$
Root an. cond. $0.954093$
Motivic weight $1$
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  − 9·5-s + 9·7-s + 8-s + 15·13-s − 9·17-s − 12·19-s + 9·23-s + 45·25-s − 27-s − 9·29-s − 9·31-s − 81·35-s + 18·37-s − 9·40-s + 27·41-s − 21·43-s + 27·47-s + 45·49-s + 9·56-s + 9·59-s − 3·61-s − 135·65-s − 3·67-s + 9·71-s − 12·73-s − 21·79-s − 9·83-s + ⋯
L(s)  = 1  − 4.02·5-s + 3.40·7-s + 0.353·8-s + 4.16·13-s − 2.18·17-s − 2.75·19-s + 1.87·23-s + 9·25-s − 0.192·27-s − 1.67·29-s − 1.61·31-s − 13.6·35-s + 2.95·37-s − 1.42·40-s + 4.21·41-s − 3.20·43-s + 3.93·47-s + 45/7·49-s + 1.20·56-s + 1.17·59-s − 0.384·61-s − 16.7·65-s − 0.366·67-s + 1.06·71-s − 1.40·73-s − 2.36·79-s − 0.987·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(0.568973\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8731996801\)
\(L(\frac12)\) \(\approx\) \(0.8731996801\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 + T^{3} + T^{6} \)
19 \( 1 + 12 T + 78 T^{2} + 385 T^{3} + 78 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 9 T + 36 T^{2} + 18 p T^{3} + 207 T^{4} + 567 T^{5} + 1441 T^{6} + 567 p T^{7} + 207 p^{2} T^{8} + 18 p^{4} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 9 T + 36 T^{2} - 115 T^{3} + 405 T^{4} - 1296 T^{5} + 3567 T^{6} - 1296 p T^{7} + 405 p^{2} T^{8} - 115 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 24 T^{2} + 18 T^{3} + 312 T^{4} - 216 T^{5} - 3593 T^{6} - 216 p T^{7} + 312 p^{2} T^{8} + 18 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 15 T + 96 T^{2} - 352 T^{3} + 963 T^{4} - 225 p T^{5} + 10521 T^{6} - 225 p^{2} T^{7} + 963 p^{2} T^{8} - 352 p^{3} T^{9} + 96 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 36 T^{2} + 72 T^{3} - 225 T^{4} - 3897 T^{5} - 21815 T^{6} - 3897 p T^{7} - 225 p^{2} T^{8} + 72 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 9 T + 36 T^{2} - 90 T^{3} + 369 T^{4} - 2025 T^{5} + 5923 T^{6} - 2025 p T^{7} + 369 p^{2} T^{8} - 90 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 9 T - 18 T^{2} - 594 T^{3} - 2331 T^{4} + 10287 T^{5} + 126217 T^{6} + 10287 p T^{7} - 2331 p^{2} T^{8} - 594 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 9 T + 54 T^{2} + 209 T^{3} - 261 T^{4} - 5958 T^{5} - 40569 T^{6} - 5958 p T^{7} - 261 p^{2} T^{8} + 209 p^{3} T^{9} + 54 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 9 T + 126 T^{2} - 665 T^{3} + 126 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 27 T + 405 T^{2} - 4473 T^{3} + 40986 T^{4} - 320760 T^{5} + 2190817 T^{6} - 320760 p T^{7} + 40986 p^{2} T^{8} - 4473 p^{3} T^{9} + 405 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 21 T + 231 T^{2} + 1979 T^{3} + 14112 T^{4} + 1962 p T^{5} + 511749 T^{6} + 1962 p^{2} T^{7} + 14112 p^{2} T^{8} + 1979 p^{3} T^{9} + 231 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 27 T + 333 T^{2} - 2439 T^{3} + 8496 T^{4} + 28422 T^{5} - 477611 T^{6} + 28422 p T^{7} + 8496 p^{2} T^{8} - 2439 p^{3} T^{9} + 333 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 90 T^{2} - 162 T^{3} + 8118 T^{4} - 15894 T^{5} + 425035 T^{6} - 15894 p T^{7} + 8118 p^{2} T^{8} - 162 p^{3} T^{9} + 90 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 - 9 T + 162 T^{2} - 1710 T^{3} + 19845 T^{4} - 153801 T^{5} + 1449937 T^{6} - 153801 p T^{7} + 19845 p^{2} T^{8} - 1710 p^{3} T^{9} + 162 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 3 T - 12 T^{2} - 388 T^{3} - 2754 T^{4} + 24057 T^{5} + 363015 T^{6} + 24057 p T^{7} - 2754 p^{2} T^{8} - 388 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T - 66 T^{2} - 982 T^{3} - 5553 T^{4} + 45963 T^{5} + 880911 T^{6} + 45963 p T^{7} - 5553 p^{2} T^{8} - 982 p^{3} T^{9} - 66 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 9 T - 27 T^{2} - 189 T^{3} + 2052 T^{4} - 16182 T^{5} + 532117 T^{6} - 16182 p T^{7} + 2052 p^{2} T^{8} - 189 p^{3} T^{9} - 27 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 12 T + 78 T^{2} + 413 T^{3} - 3663 T^{4} - 50373 T^{5} - 323559 T^{6} - 50373 p T^{7} - 3663 p^{2} T^{8} + 413 p^{3} T^{9} + 78 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 21 T + 231 T^{2} + 1223 T^{3} + 10710 T^{4} + 207918 T^{5} + 2599065 T^{6} + 207918 p T^{7} + 10710 p^{2} T^{8} + 1223 p^{3} T^{9} + 231 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 9 T - 24 T^{2} - 765 T^{3} - 5385 T^{4} - 5094 T^{5} + 401515 T^{6} - 5094 p T^{7} - 5385 p^{2} T^{8} - 765 p^{3} T^{9} - 24 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 90 T^{2} - 1503 T^{3} + 10845 T^{4} - 69993 T^{5} + 2290681 T^{6} - 69993 p T^{7} + 10845 p^{2} T^{8} - 1503 p^{3} T^{9} + 90 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 54 T + 1323 T^{2} + 18839 T^{3} + 159489 T^{4} + 678051 T^{5} + 1700166 T^{6} + 678051 p T^{7} + 159489 p^{2} T^{8} + 18839 p^{3} T^{9} + 1323 p^{4} T^{10} + 54 p^{5} T^{11} + p^{6} T^{12} \)
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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

−7.912340443219674967865210591439, −7.61847216762413594311916299717, −7.15015053932424157979659338868, −7.11096465428150460927484602150, −7.09063014282220863996841281884, −6.79548786358507246482755998169, −6.66287871492417793104828576527, −6.03434357334011024512996321350, −5.95524784806923228768015085896, −5.58106637495611631024911028452, −5.57526400348734760616258660642, −5.35393172554789353890432206348, −4.63884173917082172136083794530, −4.40107894957863005498224202632, −4.32845814376477998433386883903, −4.26451998751676803319155536322, −4.18851273398678178561434094120, −3.93929794582078748733520256717, −3.90842535542507025460507073959, −3.13064920329042396182175552699, −3.05582050226642035351720467508, −2.36041807562599896617885482882, −2.05466158097569299202966968004, −1.31991710745951298477186348114, −1.07252988782449084960696443483, 1.07252988782449084960696443483, 1.31991710745951298477186348114, 2.05466158097569299202966968004, 2.36041807562599896617885482882, 3.05582050226642035351720467508, 3.13064920329042396182175552699, 3.90842535542507025460507073959, 3.93929794582078748733520256717, 4.18851273398678178561434094120, 4.26451998751676803319155536322, 4.32845814376477998433386883903, 4.40107894957863005498224202632, 4.63884173917082172136083794530, 5.35393172554789353890432206348, 5.57526400348734760616258660642, 5.58106637495611631024911028452, 5.95524784806923228768015085896, 6.03434357334011024512996321350, 6.66287871492417793104828576527, 6.79548786358507246482755998169, 7.09063014282220863996841281884, 7.11096465428150460927484602150, 7.15015053932424157979659338868, 7.61847216762413594311916299717, 7.912340443219674967865210591439

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

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