Properties

Label 24-819e12-1.1-c1e12-0-7
Degree $24$
Conductor $9.108\times 10^{34}$
Sign $1$
Analytic cond. $6.11972\times 10^{9}$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 5-s − 3·7-s − 22·8-s − 4·10-s − 4·11-s − 2·13-s − 12·14-s − 32·16-s + 10·17-s − 19-s − 16·22-s − 2·23-s + 19·25-s − 8·26-s − 3·29-s + 16·31-s + 20·32-s + 40·34-s + 3·35-s + 26·37-s − 4·38-s + 22·40-s + 8·41-s − 11·43-s − 8·46-s + 47-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.447·5-s − 1.13·7-s − 7.77·8-s − 1.26·10-s − 1.20·11-s − 0.554·13-s − 3.20·14-s − 8·16-s + 2.42·17-s − 0.229·19-s − 3.41·22-s − 0.417·23-s + 19/5·25-s − 1.56·26-s − 0.557·29-s + 2.87·31-s + 3.53·32-s + 6.85·34-s + 0.507·35-s + 4.27·37-s − 0.648·38-s + 3.47·40-s + 1.24·41-s − 1.67·43-s − 1.17·46-s + 0.145·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(6.11972\times 10^{9}\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{24} \cdot 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.611947064\)
\(L(\frac12)\) \(\approx\) \(5.611947064\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 3 T + 6 T^{2} + 45 T^{3} + 3 p^{2} T^{4} + 348 T^{5} + 1069 T^{6} + 348 p T^{7} + 3 p^{4} T^{8} + 45 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T - 16 T^{2} + 3 T^{3} + 607 T^{4} + 433 T^{5} - 5615 T^{6} + 433 p T^{7} + 607 p^{2} T^{8} + 3 p^{3} T^{9} - 16 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
good2 \( ( 1 - p T + 3 p T^{2} - 9 T^{3} + 5 p^{2} T^{4} - 7 p^{2} T^{5} + 51 T^{6} - 7 p^{3} T^{7} + 5 p^{4} T^{8} - 9 p^{3} T^{9} + 3 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} )^{2} \)
5 \( 1 + T - 18 T^{2} + p T^{3} + 193 T^{4} - 192 T^{5} - 1181 T^{6} + 2139 T^{7} + 908 p T^{8} - 451 p^{2} T^{9} - 6679 T^{10} + 26266 T^{11} + 249 T^{12} + 26266 p T^{13} - 6679 p^{2} T^{14} - 451 p^{5} T^{15} + 908 p^{5} T^{16} + 2139 p^{5} T^{17} - 1181 p^{6} T^{18} - 192 p^{7} T^{19} + 193 p^{8} T^{20} + p^{10} T^{21} - 18 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 + 4 T - 29 T^{2} - 108 T^{3} + 477 T^{4} + 113 p T^{5} - 6686 T^{6} - 7665 T^{7} + 89323 T^{8} - 423 T^{9} - 1282040 T^{10} + 249219 T^{11} + 16505087 T^{12} + 249219 p T^{13} - 1282040 p^{2} T^{14} - 423 p^{3} T^{15} + 89323 p^{4} T^{16} - 7665 p^{5} T^{17} - 6686 p^{6} T^{18} + 113 p^{8} T^{19} + 477 p^{8} T^{20} - 108 p^{9} T^{21} - 29 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
17 \( ( 1 - 5 T + 90 T^{2} - 411 T^{3} + 3539 T^{4} - 13744 T^{5} + 78123 T^{6} - 13744 p T^{7} + 3539 p^{2} T^{8} - 411 p^{3} T^{9} + 90 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
19 \( 1 + T - 49 T^{2} + 82 T^{3} + 1336 T^{4} - 4335 T^{5} - 10907 T^{6} + 99626 T^{7} - 263580 T^{8} - 1110690 T^{9} + 9684539 T^{10} + 2414194 T^{11} - 215227743 T^{12} + 2414194 p T^{13} + 9684539 p^{2} T^{14} - 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} + 99626 p^{5} T^{17} - 10907 p^{6} T^{18} - 4335 p^{7} T^{19} + 1336 p^{8} T^{20} + 82 p^{9} T^{21} - 49 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
23 \( ( 1 + T + 32 T^{2} + 52 T^{3} + 1214 T^{4} + 1383 T^{5} + 21935 T^{6} + 1383 p T^{7} + 1214 p^{2} T^{8} + 52 p^{3} T^{9} + 32 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \)
29 \( 1 + 3 T - 3 p T^{2} - 94 T^{3} + 158 p T^{4} - 904 T^{5} - 123467 T^{6} + 308042 T^{7} + 1215550 T^{8} - 10832851 T^{9} + 73693658 T^{10} + 174959016 T^{11} - 3324017493 T^{12} + 174959016 p T^{13} + 73693658 p^{2} T^{14} - 10832851 p^{3} T^{15} + 1215550 p^{4} T^{16} + 308042 p^{5} T^{17} - 123467 p^{6} T^{18} - 904 p^{7} T^{19} + 158 p^{9} T^{20} - 94 p^{9} T^{21} - 3 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 16 T + 20 T^{2} + 594 T^{3} + 2163 T^{4} - 43649 T^{5} - 56125 T^{6} + 1282696 T^{7} + 2984747 T^{8} - 22743273 T^{9} - 180497697 T^{10} + 655302586 T^{11} + 2182678017 T^{12} + 655302586 p T^{13} - 180497697 p^{2} T^{14} - 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} + 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} - 43649 p^{7} T^{19} + 2163 p^{8} T^{20} + 594 p^{9} T^{21} + 20 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \)
37 \( ( 1 - 13 T + 184 T^{2} - 1054 T^{3} + 7158 T^{4} - 10573 T^{5} + 113729 T^{6} - 10573 p T^{7} + 7158 p^{2} T^{8} - 1054 p^{3} T^{9} + 184 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 - 8 T - 161 T^{2} + 924 T^{3} + 18241 T^{4} - 64367 T^{5} - 1502654 T^{6} + 3175261 T^{7} + 96068491 T^{8} - 105301221 T^{9} - 5078164754 T^{10} + 1647875431 T^{11} + 226350132753 T^{12} + 1647875431 p T^{13} - 5078164754 p^{2} T^{14} - 105301221 p^{3} T^{15} + 96068491 p^{4} T^{16} + 3175261 p^{5} T^{17} - 1502654 p^{6} T^{18} - 64367 p^{7} T^{19} + 18241 p^{8} T^{20} + 924 p^{9} T^{21} - 161 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 11 T - 138 T^{2} - 1349 T^{3} + 16370 T^{4} + 106653 T^{5} - 1472431 T^{6} - 5757651 T^{7} + 106708219 T^{8} + 224797058 T^{9} - 6088028976 T^{10} - 3777766292 T^{11} + 288640495545 T^{12} - 3777766292 p T^{13} - 6088028976 p^{2} T^{14} + 224797058 p^{3} T^{15} + 106708219 p^{4} T^{16} - 5757651 p^{5} T^{17} - 1472431 p^{6} T^{18} + 106653 p^{7} T^{19} + 16370 p^{8} T^{20} - 1349 p^{9} T^{21} - 138 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 - T - 104 T^{2} + 189 T^{3} + 5335 T^{4} - 164 p T^{5} - 69863 T^{6} - 514255 T^{7} - 7627520 T^{8} + 55687467 T^{9} + 662939941 T^{10} - 1686387922 T^{11} - 35399065407 T^{12} - 1686387922 p T^{13} + 662939941 p^{2} T^{14} + 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} - 514255 p^{5} T^{17} - 69863 p^{6} T^{18} - 164 p^{8} T^{19} + 5335 p^{8} T^{20} + 189 p^{9} T^{21} - 104 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 2 T - 214 T^{2} + 252 T^{3} + 24796 T^{4} - 13772 T^{5} - 1921862 T^{6} - 82142 T^{7} + 113089342 T^{8} + 43114584 T^{9} - 5653831794 T^{10} - 1443208718 T^{11} + 285781391787 T^{12} - 1443208718 p T^{13} - 5653831794 p^{2} T^{14} + 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} - 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} - 13772 p^{7} T^{19} + 24796 p^{8} T^{20} + 252 p^{9} T^{21} - 214 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
59 \( ( 1 - 13 T + 5 p T^{2} - 2839 T^{3} + 38957 T^{4} - 294699 T^{5} + 2963017 T^{6} - 294699 p T^{7} + 38957 p^{2} T^{8} - 2839 p^{3} T^{9} + 5 p^{5} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( 1 + 5 T - 140 T^{2} - 373 T^{3} + 8487 T^{4} - 5202 T^{5} - 147441 T^{6} + 963135 T^{7} - 4711566 T^{8} - 13690661 T^{9} - 1296684385 T^{10} - 689962304 T^{11} + 162150963097 T^{12} - 689962304 p T^{13} - 1296684385 p^{2} T^{14} - 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} + 963135 p^{5} T^{17} - 147441 p^{6} T^{18} - 5202 p^{7} T^{19} + 8487 p^{8} T^{20} - 373 p^{9} T^{21} - 140 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 6 T - 249 T^{2} - 278 T^{3} + 39793 T^{4} - 68141 T^{5} - 3761552 T^{6} + 15648583 T^{7} + 241594531 T^{8} - 1275513473 T^{9} - 10122739162 T^{10} + 44683203723 T^{11} + 523547364015 T^{12} + 44683203723 p T^{13} - 10122739162 p^{2} T^{14} - 1275513473 p^{3} T^{15} + 241594531 p^{4} T^{16} + 15648583 p^{5} T^{17} - 3761552 p^{6} T^{18} - 68141 p^{7} T^{19} + 39793 p^{8} T^{20} - 278 p^{9} T^{21} - 249 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 + 30 T + 224 T^{2} - 1118 T^{3} - 5021 T^{4} + 290169 T^{5} + 1854677 T^{6} - 9817892 T^{7} - 27971653 T^{8} + 688598777 T^{9} - 1819010273 T^{10} - 20701972840 T^{11} + 235631264151 T^{12} - 20701972840 p T^{13} - 1819010273 p^{2} T^{14} + 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} - 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} + 290169 p^{7} T^{19} - 5021 p^{8} T^{20} - 1118 p^{9} T^{21} + 224 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \)
83 \( ( 1 - 27 T + 656 T^{2} - 10802 T^{3} + 153994 T^{4} - 1760871 T^{5} + 17670883 T^{6} - 1760871 p T^{7} + 153994 p^{2} T^{8} - 10802 p^{3} T^{9} + 656 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 4 T + 167 T^{2} - 1648 T^{3} + 21035 T^{4} - 202110 T^{5} + 2204075 T^{6} - 202110 p T^{7} + 21035 p^{2} T^{8} - 1648 p^{3} T^{9} + 167 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 + 35 T + 278 T^{2} - 3177 T^{3} - 20496 T^{4} + 1111333 T^{5} + 13328183 T^{6} - 54713297 T^{7} - 1182920923 T^{8} + 11775176076 T^{9} + 251445486222 T^{10} + 186060844192 T^{11} - 17274836413101 T^{12} + 186060844192 p T^{13} + 251445486222 p^{2} T^{14} + 11775176076 p^{3} T^{15} - 1182920923 p^{4} T^{16} - 54713297 p^{5} T^{17} + 13328183 p^{6} T^{18} + 1111333 p^{7} T^{19} - 20496 p^{8} T^{20} - 3177 p^{9} T^{21} + 278 p^{10} T^{22} + 35 p^{11} T^{23} + p^{12} T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.14719249853650921743529776739, −3.13765727424324796419033677463, −3.11206157759913882371155127545, −3.10057409972799192504338787962, −3.05103233645760830359897541282, −3.04723527983199537025488103579, −2.88095485010120275299664620678, −2.73557202657388232337604905395, −2.63222610059935418708890490608, −2.52063404593312267626766627345, −2.47314901423789411542281382915, −2.11798499969535451287421936931, −2.04021813421898178406133816781, −2.03909257385945414255947209010, −2.02138691172068749976143588922, −1.99350922290984693771327939166, −1.41567818263652864687905853235, −1.29989463111603649165655290291, −0.959570804471549353224845178751, −0.951515938657172648325185370459, −0.945389770868697198052926512062, −0.924449515598442546240690064625, −0.61492725103183307822837622676, −0.37595699925707566388206069861, −0.21772789153123516617163295682, 0.21772789153123516617163295682, 0.37595699925707566388206069861, 0.61492725103183307822837622676, 0.924449515598442546240690064625, 0.945389770868697198052926512062, 0.951515938657172648325185370459, 0.959570804471549353224845178751, 1.29989463111603649165655290291, 1.41567818263652864687905853235, 1.99350922290984693771327939166, 2.02138691172068749976143588922, 2.03909257385945414255947209010, 2.04021813421898178406133816781, 2.11798499969535451287421936931, 2.47314901423789411542281382915, 2.52063404593312267626766627345, 2.63222610059935418708890490608, 2.73557202657388232337604905395, 2.88095485010120275299664620678, 3.04723527983199537025488103579, 3.05103233645760830359897541282, 3.10057409972799192504338787962, 3.11206157759913882371155127545, 3.13765727424324796419033677463, 3.14719249853650921743529776739

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

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