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Properties

Label 12-912e6-1.1-c1e6-0-7
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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 + 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 + 27·41-s + 21·43-s − 27·47-s + 45·49-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 + 81·85-s − 135·91-s − 108·95-s + ⋯
L(s)  = 1  − 4.02·5-s − 3.40·7-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 + 4.21·41-s + 3.20·43-s − 3.93·47-s + 45/7·49-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 + 8.78·85-s − 14.1·91-s − 11.0·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6433143151\)
\(L(\frac12)\) \(\approx\) \(0.6433143151\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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

−5.39967963612381948913983393683, −5.28831906542665047636203646489, −4.71832133322812277762997933852, −4.66652901032970216527205831707, −4.61221837863720717239391262922, −4.45510023528540704991205765000, −4.21940048666628694139665427698, −3.90250401537310624133100983712, −3.85686166189794910304913331572, −3.81039118633239094143998790805, −3.78859541369295844704875262870, −3.66984554591541632963818929672, −3.50748089575115992552806464806, −3.02966724378577809872720063017, −2.92743298205757906107499556463, −2.84139849463979899835749015910, −2.68680374396991678373623382566, −2.67503870215504840375059143904, −2.24236727518860253178100418039, −1.63146808079526719273960050037, −1.23838015324240502164794760416, −1.11320082895304254423362066559, −0.924187222742347068270676713545, −0.41462179744505619210083994590, −0.32762515712489545700537373527, 0.32762515712489545700537373527, 0.41462179744505619210083994590, 0.924187222742347068270676713545, 1.11320082895304254423362066559, 1.23838015324240502164794760416, 1.63146808079526719273960050037, 2.24236727518860253178100418039, 2.67503870215504840375059143904, 2.68680374396991678373623382566, 2.84139849463979899835749015910, 2.92743298205757906107499556463, 3.02966724378577809872720063017, 3.50748089575115992552806464806, 3.66984554591541632963818929672, 3.78859541369295844704875262870, 3.81039118633239094143998790805, 3.85686166189794910304913331572, 3.90250401537310624133100983712, 4.21940048666628694139665427698, 4.45510023528540704991205765000, 4.61221837863720717239391262922, 4.66652901032970216527205831707, 4.71832133322812277762997933852, 5.28831906542665047636203646489, 5.39967963612381948913983393683

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

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