Properties

Label 24-3e60-1.1-c1e12-0-3
Degree $24$
Conductor $4.239\times 10^{28}$
Sign $1$
Analytic cond. $2848.39$
Root an. cond. $1.39296$
Motivic weight $1$
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  + 3·2-s + 6·4-s − 3·5-s + 3·7-s + 9·8-s − 9·10-s + 3·11-s + 3·13-s + 9·14-s + 12·16-s + 9·17-s − 3·19-s − 18·20-s + 9·22-s + 24·23-s − 3·25-s + 9·26-s + 18·28-s + 30·29-s − 15·31-s + 3·32-s + 27·34-s − 9·35-s − 3·37-s − 9·38-s − 27·40-s − 21·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s − 1.34·5-s + 1.13·7-s + 3.18·8-s − 2.84·10-s + 0.904·11-s + 0.832·13-s + 2.40·14-s + 3·16-s + 2.18·17-s − 0.688·19-s − 4.02·20-s + 1.91·22-s + 5.00·23-s − 3/5·25-s + 1.76·26-s + 3.40·28-s + 5.57·29-s − 2.69·31-s + 0.530·32-s + 4.63·34-s − 1.52·35-s − 0.493·37-s − 1.45·38-s − 4.26·40-s − 3.27·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{60}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{60}\)
Sign: $1$
Analytic conductor: \(2848.39\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{60} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(23.27432163\)
\(L(\frac12)\) \(\approx\) \(23.27432163\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( ( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}( 1 + 3 T - 9 T^{3} - 9 T^{4} + 3 p^{2} T^{5} + 37 T^{6} + 3 p^{3} T^{7} - 9 p^{2} T^{8} - 9 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} ) \)
5 \( 1 + 3 T + 12 T^{2} + 9 p T^{3} + 132 T^{4} + 327 T^{5} + 971 T^{6} + 2187 T^{7} + 4878 T^{8} + 11367 T^{9} + 24048 T^{10} + 48969 T^{11} + 116649 T^{12} + 48969 p T^{13} + 24048 p^{2} T^{14} + 11367 p^{3} T^{15} + 4878 p^{4} T^{16} + 2187 p^{5} T^{17} + 971 p^{6} T^{18} + 327 p^{7} T^{19} + 132 p^{8} T^{20} + 9 p^{10} T^{21} + 12 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 - 3 T - 6 T^{2} + 16 T^{3} + 93 T^{4} - 255 T^{5} - 305 T^{6} + 1620 T^{7} + 243 p T^{8} - 5645 T^{9} - 3837 p T^{10} + 55605 T^{11} + 48757 T^{12} + 55605 p T^{13} - 3837 p^{3} T^{14} - 5645 p^{3} T^{15} + 243 p^{5} T^{16} + 1620 p^{5} T^{17} - 305 p^{6} T^{18} - 255 p^{7} T^{19} + 93 p^{8} T^{20} + 16 p^{9} T^{21} - 6 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 3 T + 21 T^{2} + 18 T^{3} + 141 T^{4} + 141 T^{5} + 3662 T^{6} - 783 T^{7} + 5283 T^{8} + 101736 T^{9} - 200457 T^{10} + 203103 T^{11} + 383733 T^{12} + 203103 p T^{13} - 200457 p^{2} T^{14} + 101736 p^{3} T^{15} + 5283 p^{4} T^{16} - 783 p^{5} T^{17} + 3662 p^{6} T^{18} + 141 p^{7} T^{19} + 141 p^{8} T^{20} + 18 p^{9} T^{21} + 21 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 3 T + 3 T^{2} - 2 T^{3} - 447 T^{4} + 1383 T^{5} - 1529 T^{6} - 1674 T^{7} + 110538 T^{8} - 221726 T^{9} + 196566 T^{10} + 1331472 T^{11} - 20709779 T^{12} + 1331472 p T^{13} + 196566 p^{2} T^{14} - 221726 p^{3} T^{15} + 110538 p^{4} T^{16} - 1674 p^{5} T^{17} - 1529 p^{6} T^{18} + 1383 p^{7} T^{19} - 447 p^{8} T^{20} - 2 p^{9} T^{21} + 3 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 9 T - 30 T^{2} + 423 T^{3} + 1029 T^{4} - 14184 T^{5} - 23521 T^{6} + 296649 T^{7} + 637560 T^{8} - 4620213 T^{9} - 12537675 T^{10} + 28264410 T^{11} + 250681641 T^{12} + 28264410 p T^{13} - 12537675 p^{2} T^{14} - 4620213 p^{3} T^{15} + 637560 p^{4} T^{16} + 296649 p^{5} T^{17} - 23521 p^{6} T^{18} - 14184 p^{7} T^{19} + 1029 p^{8} T^{20} + 423 p^{9} T^{21} - 30 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 3 T - 75 T^{2} - 242 T^{3} + 3012 T^{4} + 9714 T^{5} - 85589 T^{6} - 257166 T^{7} + 1946502 T^{8} + 4391737 T^{9} - 39399504 T^{10} - 1762662 p T^{11} + 40166287 p T^{12} - 1762662 p^{2} T^{13} - 39399504 p^{2} T^{14} + 4391737 p^{3} T^{15} + 1946502 p^{4} T^{16} - 257166 p^{5} T^{17} - 85589 p^{6} T^{18} + 9714 p^{7} T^{19} + 3012 p^{8} T^{20} - 242 p^{9} T^{21} - 75 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 - 24 T + 246 T^{2} - 1143 T^{3} - 372 T^{4} + 31170 T^{5} - 111106 T^{6} - 289530 T^{7} + 2718054 T^{8} - 556335 T^{9} - 22916826 T^{10} - 237389940 T^{11} + 2410995357 T^{12} - 237389940 p T^{13} - 22916826 p^{2} T^{14} - 556335 p^{3} T^{15} + 2718054 p^{4} T^{16} - 289530 p^{5} T^{17} - 111106 p^{6} T^{18} + 31170 p^{7} T^{19} - 372 p^{8} T^{20} - 1143 p^{9} T^{21} + 246 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 30 T + 408 T^{2} - 3042 T^{3} + 336 p T^{4} + 43296 T^{5} - 688807 T^{6} + 3526866 T^{7} - 3801609 T^{8} - 62079696 T^{9} + 385906887 T^{10} - 559725588 T^{11} - 2188204731 T^{12} - 559725588 p T^{13} + 385906887 p^{2} T^{14} - 62079696 p^{3} T^{15} - 3801609 p^{4} T^{16} + 3526866 p^{5} T^{17} - 688807 p^{6} T^{18} + 43296 p^{7} T^{19} + 336 p^{9} T^{20} - 3042 p^{9} T^{21} + 408 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 + 15 T + 3 p T^{2} + 79 T^{3} - 4947 T^{4} - 46659 T^{5} - 180971 T^{6} + 344871 T^{7} + 10274067 T^{8} + 61371322 T^{9} + 97258128 T^{10} - 1342000518 T^{11} - 12908637923 T^{12} - 1342000518 p T^{13} + 97258128 p^{2} T^{14} + 61371322 p^{3} T^{15} + 10274067 p^{4} T^{16} + 344871 p^{5} T^{17} - 180971 p^{6} T^{18} - 46659 p^{7} T^{19} - 4947 p^{8} T^{20} + 79 p^{9} T^{21} + 3 p^{11} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 + 3 T - 156 T^{2} - 107 T^{3} + 13731 T^{4} - 9132 T^{5} - 864755 T^{6} + 641043 T^{7} + 43249536 T^{8} - 18536771 T^{9} - 1953626739 T^{10} + 269355786 T^{11} + 78884071369 T^{12} + 269355786 p T^{13} - 1953626739 p^{2} T^{14} - 18536771 p^{3} T^{15} + 43249536 p^{4} T^{16} + 641043 p^{5} T^{17} - 864755 p^{6} T^{18} - 9132 p^{7} T^{19} + 13731 p^{8} T^{20} - 107 p^{9} T^{21} - 156 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 21 T + 264 T^{2} + 2637 T^{3} + 22560 T^{4} + 160689 T^{5} + 955475 T^{6} + 4208229 T^{7} + 2621097 T^{8} - 180841572 T^{9} - 2278154898 T^{10} - 19816693596 T^{11} - 139816422693 T^{12} - 19816693596 p T^{13} - 2278154898 p^{2} T^{14} - 180841572 p^{3} T^{15} + 2621097 p^{4} T^{16} + 4208229 p^{5} T^{17} + 955475 p^{6} T^{18} + 160689 p^{7} T^{19} + 22560 p^{8} T^{20} + 2637 p^{9} T^{21} + 264 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 12 T + 111 T^{2} - 929 T^{3} + 9930 T^{4} - 1920 p T^{5} + 694621 T^{6} - 5307525 T^{7} + 40383765 T^{8} - 286082336 T^{9} + 2151896109 T^{10} - 14454219750 T^{11} + 96620287315 T^{12} - 14454219750 p T^{13} + 2151896109 p^{2} T^{14} - 286082336 p^{3} T^{15} + 40383765 p^{4} T^{16} - 5307525 p^{5} T^{17} + 694621 p^{6} T^{18} - 1920 p^{8} T^{19} + 9930 p^{8} T^{20} - 929 p^{9} T^{21} + 111 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 - 3 T - 69 T^{2} + 549 T^{3} + 879 T^{4} - 46686 T^{5} + 146141 T^{6} + 1585800 T^{7} - 11836998 T^{8} - 26147259 T^{9} + 625936104 T^{10} + 44162253 T^{11} - 25947595719 T^{12} + 44162253 p T^{13} + 625936104 p^{2} T^{14} - 26147259 p^{3} T^{15} - 11836998 p^{4} T^{16} + 1585800 p^{5} T^{17} + 146141 p^{6} T^{18} - 46686 p^{7} T^{19} + 879 p^{8} T^{20} + 549 p^{9} T^{21} - 69 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
53 \( ( 1 + 9 T + 210 T^{2} + 1872 T^{3} + 23856 T^{4} + 168327 T^{5} + 1634317 T^{6} + 168327 p T^{7} + 23856 p^{2} T^{8} + 1872 p^{3} T^{9} + 210 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
59 \( 1 - 15 T + 12 T^{2} + 1566 T^{3} - 20523 T^{4} + 148053 T^{5} - 36559 T^{6} - 13518252 T^{7} + 176997411 T^{8} - 1060478919 T^{9} - 492616251 T^{10} + 71985657519 T^{11} - 747587755359 T^{12} + 71985657519 p T^{13} - 492616251 p^{2} T^{14} - 1060478919 p^{3} T^{15} + 176997411 p^{4} T^{16} - 13518252 p^{5} T^{17} - 36559 p^{6} T^{18} + 148053 p^{7} T^{19} - 20523 p^{8} T^{20} + 1566 p^{9} T^{21} + 12 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 21 T + 255 T^{2} - 2873 T^{3} + 31674 T^{4} - 325488 T^{5} + 2991511 T^{6} - 25168941 T^{7} + 205375149 T^{8} - 1590079142 T^{9} + 11979788397 T^{10} - 93473672865 T^{11} + 731800605235 T^{12} - 93473672865 p T^{13} + 11979788397 p^{2} T^{14} - 1590079142 p^{3} T^{15} + 205375149 p^{4} T^{16} - 25168941 p^{5} T^{17} + 2991511 p^{6} T^{18} - 325488 p^{7} T^{19} + 31674 p^{8} T^{20} - 2873 p^{9} T^{21} + 255 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 - 21 T + 102 T^{2} + 1213 T^{3} - 29292 T^{4} + 278277 T^{5} - 290051 T^{6} - 19466865 T^{7} + 243875907 T^{8} - 1602601148 T^{9} - 515624682 T^{10} + 114363308010 T^{11} - 1211418447647 T^{12} + 114363308010 p T^{13} - 515624682 p^{2} T^{14} - 1602601148 p^{3} T^{15} + 243875907 p^{4} T^{16} - 19466865 p^{5} T^{17} - 290051 p^{6} T^{18} + 278277 p^{7} T^{19} - 29292 p^{8} T^{20} + 1213 p^{9} T^{21} + 102 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 27 T + 78 T^{2} + 2565 T^{3} + 13071 T^{4} - 524664 T^{5} - 751711 T^{6} + 30297321 T^{7} + 410765508 T^{8} - 3391054713 T^{9} - 30034133541 T^{10} + 13624108308 T^{11} + 3600759258249 T^{12} + 13624108308 p T^{13} - 30034133541 p^{2} T^{14} - 3391054713 p^{3} T^{15} + 410765508 p^{4} T^{16} + 30297321 p^{5} T^{17} - 751711 p^{6} T^{18} - 524664 p^{7} T^{19} + 13071 p^{8} T^{20} + 2565 p^{9} T^{21} + 78 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 6 T - 228 T^{2} + 2296 T^{3} + 24945 T^{4} - 381255 T^{5} - 980072 T^{6} + 40200363 T^{7} - 102286134 T^{8} - 2648934335 T^{9} + 21743689350 T^{10} + 78452536893 T^{11} - 2017821540323 T^{12} + 78452536893 p T^{13} + 21743689350 p^{2} T^{14} - 2648934335 p^{3} T^{15} - 102286134 p^{4} T^{16} + 40200363 p^{5} T^{17} - 980072 p^{6} T^{18} - 381255 p^{7} T^{19} + 24945 p^{8} T^{20} + 2296 p^{9} T^{21} - 228 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 21 T + 255 T^{2} - 2198 T^{3} + 16959 T^{4} - 110838 T^{5} + 526249 T^{6} + 2937060 T^{7} - 43672446 T^{8} - 259698530 T^{9} + 7366588728 T^{10} - 78109520181 T^{11} + 690348673255 T^{12} - 78109520181 p T^{13} + 7366588728 p^{2} T^{14} - 259698530 p^{3} T^{15} - 43672446 p^{4} T^{16} + 2937060 p^{5} T^{17} + 526249 p^{6} T^{18} - 110838 p^{7} T^{19} + 16959 p^{8} T^{20} - 2198 p^{9} T^{21} + 255 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 33 T + 597 T^{2} + 7317 T^{3} + 58749 T^{4} + 284199 T^{5} + 392219 T^{6} + 2919609 T^{7} + 216160263 T^{8} + 3986032086 T^{9} + 45502498566 T^{10} + 398032175520 T^{11} + 3390402467853 T^{12} + 398032175520 p T^{13} + 45502498566 p^{2} T^{14} + 3986032086 p^{3} T^{15} + 216160263 p^{4} T^{16} + 2919609 p^{5} T^{17} + 392219 p^{6} T^{18} + 284199 p^{7} T^{19} + 58749 p^{8} T^{20} + 7317 p^{9} T^{21} + 597 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 - 9 T - 273 T^{2} + 2772 T^{3} + 38802 T^{4} - 449316 T^{5} - 3561871 T^{6} + 54551502 T^{7} + 157767516 T^{8} - 4371660207 T^{9} + 3816883044 T^{10} + 152630961444 T^{11} - 900621732009 T^{12} + 152630961444 p T^{13} + 3816883044 p^{2} T^{14} - 4371660207 p^{3} T^{15} + 157767516 p^{4} T^{16} + 54551502 p^{5} T^{17} - 3561871 p^{6} T^{18} - 449316 p^{7} T^{19} + 38802 p^{8} T^{20} + 2772 p^{9} T^{21} - 273 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 42 T + 921 T^{2} + 13273 T^{3} + 134454 T^{4} + 989178 T^{5} + 5788711 T^{6} + 55187271 T^{7} + 970885899 T^{8} + 14851895560 T^{9} + 171553448337 T^{10} + 1647392209770 T^{11} + 15680763033367 T^{12} + 1647392209770 p T^{13} + 171553448337 p^{2} T^{14} + 14851895560 p^{3} T^{15} + 970885899 p^{4} T^{16} + 55187271 p^{5} T^{17} + 5788711 p^{6} T^{18} + 989178 p^{7} T^{19} + 134454 p^{8} T^{20} + 13273 p^{9} T^{21} + 921 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.03081770294757179022462841406, −3.92314073856413809031403883548, −3.91690869741492047629805723770, −3.82818342106099149079244229655, −3.80001964584520085883526042916, −3.79014165510334155319194330646, −3.71151217403148752255887640153, −3.51753279989395852423166052608, −3.25037781159288913869873602473, −3.22545817610469367938676014630, −2.97167219110954363888102359523, −2.92083756810278092988801130617, −2.75500781882774089626901301080, −2.68966230909476029444390099888, −2.68574515240628302967933472131, −2.53929463863001676388537566997, −2.15778325942389840041649272602, −2.13863669683817358063535052656, −1.80722451347292545381128242095, −1.65227344613671298219893064866, −1.43961061496835071701722404354, −1.39107666701908968542744660339, −1.03320269830058611252362954666, −0.852786274537696414959674656872, −0.837031408139444010986086460616, 0.837031408139444010986086460616, 0.852786274537696414959674656872, 1.03320269830058611252362954666, 1.39107666701908968542744660339, 1.43961061496835071701722404354, 1.65227344613671298219893064866, 1.80722451347292545381128242095, 2.13863669683817358063535052656, 2.15778325942389840041649272602, 2.53929463863001676388537566997, 2.68574515240628302967933472131, 2.68966230909476029444390099888, 2.75500781882774089626901301080, 2.92083756810278092988801130617, 2.97167219110954363888102359523, 3.22545817610469367938676014630, 3.25037781159288913869873602473, 3.51753279989395852423166052608, 3.71151217403148752255887640153, 3.79014165510334155319194330646, 3.80001964584520085883526042916, 3.82818342106099149079244229655, 3.91690869741492047629805723770, 3.92314073856413809031403883548, 4.03081770294757179022462841406

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

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