Properties

Label 12-1859e6-1.1-c1e6-0-2
Degree $12$
Conductor $4.127\times 10^{19}$
Sign $1$
Analytic cond. $1.06988\times 10^{7}$
Root an. cond. $3.85281$
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  + 3-s − 2·4-s + 6·5-s + 3·7-s − 8-s − 5·9-s + 6·11-s − 2·12-s + 6·15-s + 2·17-s + 10·19-s − 12·20-s + 3·21-s + 3·23-s − 24-s − 3·27-s − 6·28-s + 3·29-s + 5·31-s + 3·32-s + 6·33-s + 18·35-s + 10·36-s + 25·37-s − 6·40-s + 24·41-s − 8·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 2.68·5-s + 1.13·7-s − 0.353·8-s − 5/3·9-s + 1.80·11-s − 0.577·12-s + 1.54·15-s + 0.485·17-s + 2.29·19-s − 2.68·20-s + 0.654·21-s + 0.625·23-s − 0.204·24-s − 0.577·27-s − 1.13·28-s + 0.557·29-s + 0.898·31-s + 0.530·32-s + 1.04·33-s + 3.04·35-s + 5/3·36-s + 4.10·37-s − 0.948·40-s + 3.74·41-s − 1.21·43-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(11^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(1.06988\times 10^{7}\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 11^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(25.81480827\)
\(L(\frac12)\) \(\approx\) \(25.81480827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 - T )^{6} \)
13 \( 1 \)
good2 \( 1 + p T^{2} + T^{3} + p^{2} T^{4} + T^{5} + 13 T^{6} + p T^{7} + p^{4} T^{8} + p^{3} T^{9} + p^{5} T^{10} + p^{6} T^{12} \)
3 \( 1 - T + 2 p T^{2} - 8 T^{3} + 17 T^{4} - 13 T^{5} + 50 T^{6} - 13 p T^{7} + 17 p^{2} T^{8} - 8 p^{3} T^{9} + 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 6 T + 36 T^{2} - 27 p T^{3} + 481 T^{4} - 1288 T^{5} + 3259 T^{6} - 1288 p T^{7} + 481 p^{2} T^{8} - 27 p^{4} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 3 T + 26 T^{2} - 78 T^{3} + 369 T^{4} - 955 T^{5} + 3182 T^{6} - 955 p T^{7} + 369 p^{2} T^{8} - 78 p^{3} T^{9} + 26 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 2 T + 4 p T^{2} - 157 T^{3} + 2283 T^{4} - 4926 T^{5} + 48005 T^{6} - 4926 p T^{7} + 2283 p^{2} T^{8} - 157 p^{3} T^{9} + 4 p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 10 T + 105 T^{2} - 739 T^{3} + 4773 T^{4} - 25107 T^{5} + 118980 T^{6} - 25107 p T^{7} + 4773 p^{2} T^{8} - 739 p^{3} T^{9} + 105 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 3 T + 82 T^{2} - 330 T^{3} + 3687 T^{4} - 13099 T^{5} + 107772 T^{6} - 13099 p T^{7} + 3687 p^{2} T^{8} - 330 p^{3} T^{9} + 82 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 3 T + 48 T^{2} - 238 T^{3} + 1699 T^{4} - 8619 T^{5} + 46693 T^{6} - 8619 p T^{7} + 1699 p^{2} T^{8} - 238 p^{3} T^{9} + 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 5 T + 5 p T^{2} - 718 T^{3} + 10921 T^{4} - 42369 T^{5} + 438894 T^{6} - 42369 p T^{7} + 10921 p^{2} T^{8} - 718 p^{3} T^{9} + 5 p^{5} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 25 T + 398 T^{2} - 4544 T^{3} + 42293 T^{4} - 323915 T^{5} + 2136093 T^{6} - 323915 p T^{7} + 42293 p^{2} T^{8} - 4544 p^{3} T^{9} + 398 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 24 T + 428 T^{2} - 5235 T^{3} + 53421 T^{4} - 435470 T^{5} + 3073001 T^{6} - 435470 p T^{7} + 53421 p^{2} T^{8} - 5235 p^{3} T^{9} + 428 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 8 T + 5 p T^{2} + 1411 T^{3} + 20885 T^{4} + 110327 T^{5} + 1161502 T^{6} + 110327 p T^{7} + 20885 p^{2} T^{8} + 1411 p^{3} T^{9} + 5 p^{5} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 10 T + 267 T^{2} - 2167 T^{3} + 30479 T^{4} - 4151 p T^{5} + 1893064 T^{6} - 4151 p^{2} T^{7} + 30479 p^{2} T^{8} - 2167 p^{3} T^{9} + 267 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 10 T + 162 T^{2} - 653 T^{3} + 4413 T^{4} + 37800 T^{5} - 144463 T^{6} + 37800 p T^{7} + 4413 p^{2} T^{8} - 653 p^{3} T^{9} + 162 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 4 T + 205 T^{2} + 711 T^{3} + 22843 T^{4} + 68723 T^{5} + 1660624 T^{6} + 68723 p T^{7} + 22843 p^{2} T^{8} + 711 p^{3} T^{9} + 205 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 21 T + 510 T^{2} + 6648 T^{3} + 89941 T^{4} + 822629 T^{5} + 7630689 T^{6} + 822629 p T^{7} + 89941 p^{2} T^{8} + 6648 p^{3} T^{9} + 510 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 21 T + 496 T^{2} - 6644 T^{3} + 90385 T^{4} - 867365 T^{5} + 8263006 T^{6} - 867365 p T^{7} + 90385 p^{2} T^{8} - 6644 p^{3} T^{9} + 496 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 3 T + 154 T^{2} + 154 T^{3} + 13897 T^{4} - 16993 T^{5} + 1035312 T^{6} - 16993 p T^{7} + 13897 p^{2} T^{8} + 154 p^{3} T^{9} + 154 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 13 T + 487 T^{2} - 4750 T^{3} + 93989 T^{4} - 693669 T^{5} + 9310278 T^{6} - 693669 p T^{7} + 93989 p^{2} T^{8} - 4750 p^{3} T^{9} + 487 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 4 T + 66 T^{2} + 546 T^{3} + 13463 T^{4} + 41230 T^{5} + 717404 T^{6} + 41230 p T^{7} + 13463 p^{2} T^{8} + 546 p^{3} T^{9} + 66 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 8 T + 321 T^{2} - 2923 T^{3} + 51931 T^{4} - 440473 T^{5} + 5329268 T^{6} - 440473 p T^{7} + 51931 p^{2} T^{8} - 2923 p^{3} T^{9} + 321 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 9 T + 206 T^{2} + 306 T^{3} + 19253 T^{4} + 14065 T^{5} + 2221884 T^{6} + 14065 p T^{7} + 19253 p^{2} T^{8} + 306 p^{3} T^{9} + 206 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 15 T + 315 T^{2} - 4184 T^{3} + 46829 T^{4} - 527309 T^{5} + 4974874 T^{6} - 527309 p T^{7} + 46829 p^{2} T^{8} - 4184 p^{3} T^{9} + 315 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.78152998866844945762412677408, −4.62706183476986934868352690453, −4.57937500107288366905604754061, −4.32039115993984026991395554841, −4.27581700315644906212730109770, −4.15173575778830129852472708326, −4.03835169220121623015155173401, −3.61191153303416079666925451764, −3.48793332345944304098925263779, −3.36080805999743383742397616908, −3.22745875259817960855736685905, −3.14869898770133128956153333974, −2.75184271072065606385184733640, −2.63532502826337060406351140144, −2.51020549194201023627694971842, −2.40386701631716552711210165179, −2.30868402080580209624655234171, −1.87665110420782801129142826713, −1.76500988075011739284750518644, −1.60150341296260810146267138340, −1.43843308753015669666570131716, −1.10959997272579234826762890770, −0.71288728839891128672135862686, −0.69939782425892569239655156314, −0.61453258534102479974295723372, 0.61453258534102479974295723372, 0.69939782425892569239655156314, 0.71288728839891128672135862686, 1.10959997272579234826762890770, 1.43843308753015669666570131716, 1.60150341296260810146267138340, 1.76500988075011739284750518644, 1.87665110420782801129142826713, 2.30868402080580209624655234171, 2.40386701631716552711210165179, 2.51020549194201023627694971842, 2.63532502826337060406351140144, 2.75184271072065606385184733640, 3.14869898770133128956153333974, 3.22745875259817960855736685905, 3.36080805999743383742397616908, 3.48793332345944304098925263779, 3.61191153303416079666925451764, 4.03835169220121623015155173401, 4.15173575778830129852472708326, 4.27581700315644906212730109770, 4.32039115993984026991395554841, 4.57937500107288366905604754061, 4.62706183476986934868352690453, 4.78152998866844945762412677408

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

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