Properties

Label 24-20e24-1.1-c1e12-0-3
Degree $24$
Conductor $1.678\times 10^{31}$
Sign $1$
Analytic cond. $1.12731\times 10^{6}$
Root an. cond. $1.78718$
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  − 2·3-s − 4·5-s + 2·7-s + 5·9-s + 5·11-s − 2·13-s + 8·15-s + 17-s + 8·19-s − 4·21-s + 6·23-s − 5·25-s + 27-s − 18·29-s − 12·31-s − 10·33-s − 8·35-s + 13·37-s + 4·39-s − 23·41-s − 50·43-s − 20·45-s − 47-s − 23·49-s − 2·51-s + 21·53-s − 20·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 0.755·7-s + 5/3·9-s + 1.50·11-s − 0.554·13-s + 2.06·15-s + 0.242·17-s + 1.83·19-s − 0.872·21-s + 1.25·23-s − 25-s + 0.192·27-s − 3.34·29-s − 2.15·31-s − 1.74·33-s − 1.35·35-s + 2.13·37-s + 0.640·39-s − 3.59·41-s − 7.62·43-s − 2.98·45-s − 0.145·47-s − 3.28·49-s − 0.280·51-s + 2.88·53-s − 2.69·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(1.12731\times 10^{6}\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.561128213\)
\(L(\frac12)\) \(\approx\) \(1.561128213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 4 T + 21 T^{2} + 74 T^{3} + 226 T^{4} + 126 p T^{5} + 288 p T^{6} + 126 p^{2} T^{7} + 226 p^{2} T^{8} + 74 p^{3} T^{9} + 21 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
good3 \( 1 + 2 T - T^{2} - 13 T^{3} - 28 T^{4} - 29 T^{5} + 19 T^{6} + 194 T^{7} + 455 T^{8} + 583 T^{9} - 49 T^{10} - 2153 T^{11} - 5330 T^{12} - 2153 p T^{13} - 49 p^{2} T^{14} + 583 p^{3} T^{15} + 455 p^{4} T^{16} + 194 p^{5} T^{17} + 19 p^{6} T^{18} - 29 p^{7} T^{19} - 28 p^{8} T^{20} - 13 p^{9} T^{21} - p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
7 \( ( 1 - T + 13 T^{2} - 17 T^{3} + 167 T^{4} - 208 T^{5} + 1174 T^{6} - 208 p T^{7} + 167 p^{2} T^{8} - 17 p^{3} T^{9} + 13 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( 1 - 5 T + 7 T^{2} - 65 T^{3} + 351 T^{4} - 1490 T^{5} + 6090 T^{6} - 16160 T^{7} + 72390 T^{8} - 311790 T^{9} + 1151332 T^{10} - 3608920 T^{11} + 9338114 T^{12} - 3608920 p T^{13} + 1151332 p^{2} T^{14} - 311790 p^{3} T^{15} + 72390 p^{4} T^{16} - 16160 p^{5} T^{17} + 6090 p^{6} T^{18} - 1490 p^{7} T^{19} + 351 p^{8} T^{20} - 65 p^{9} T^{21} + 7 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 + 2 T - 16 T^{2} + 2 T^{3} - 53 T^{4} + 6 T^{5} + 7544 T^{6} + 1784 T^{7} - 56280 T^{8} + 53458 T^{9} - 477164 T^{10} + 428612 T^{11} + 19743920 T^{12} + 428612 p T^{13} - 477164 p^{2} T^{14} + 53458 p^{3} T^{15} - 56280 p^{4} T^{16} + 1784 p^{5} T^{17} + 7544 p^{6} T^{18} + 6 p^{7} T^{19} - 53 p^{8} T^{20} + 2 p^{9} T^{21} - 16 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - T - 9 T^{2} + 3 p T^{3} + 507 T^{4} - 718 T^{5} - 6864 T^{6} + 9252 T^{7} + 172520 T^{8} - 281314 T^{9} - 3362156 T^{10} - 273914 T^{11} + 40433450 T^{12} - 273914 p T^{13} - 3362156 p^{2} T^{14} - 281314 p^{3} T^{15} + 172520 p^{4} T^{16} + 9252 p^{5} T^{17} - 6864 p^{6} T^{18} - 718 p^{7} T^{19} + 507 p^{8} T^{20} + 3 p^{10} T^{21} - 9 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 - 8 T - 6 T^{2} + 258 T^{3} - 939 T^{4} + 1272 T^{5} + 8744 T^{6} - 96932 T^{7} + 222296 T^{8} + 392992 T^{9} + 160794 T^{10} + 5760978 T^{11} - 96272964 T^{12} + 5760978 p T^{13} + 160794 p^{2} T^{14} + 392992 p^{3} T^{15} + 222296 p^{4} T^{16} - 96932 p^{5} T^{17} + 8744 p^{6} T^{18} + 1272 p^{7} T^{19} - 939 p^{8} T^{20} + 258 p^{9} T^{21} - 6 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 - 6 T + 13 T^{2} + 155 T^{3} - 428 T^{4} + 3269 T^{5} - 9845 T^{6} + 111568 T^{7} + 95053 T^{8} + 1251629 T^{9} + 5100635 T^{10} - 36361407 T^{11} + 637218038 T^{12} - 36361407 p T^{13} + 5100635 p^{2} T^{14} + 1251629 p^{3} T^{15} + 95053 p^{4} T^{16} + 111568 p^{5} T^{17} - 9845 p^{6} T^{18} + 3269 p^{7} T^{19} - 428 p^{8} T^{20} + 155 p^{9} T^{21} + 13 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 + 18 T + 154 T^{2} + 1212 T^{3} + 10291 T^{4} + 77478 T^{5} + 547974 T^{6} + 3752112 T^{7} + 23821306 T^{8} + 144955698 T^{9} + 863261644 T^{10} + 4979066562 T^{11} + 27518493276 T^{12} + 4979066562 p T^{13} + 863261644 p^{2} T^{14} + 144955698 p^{3} T^{15} + 23821306 p^{4} T^{16} + 3752112 p^{5} T^{17} + 547974 p^{6} T^{18} + 77478 p^{7} T^{19} + 10291 p^{8} T^{20} + 1212 p^{9} T^{21} + 154 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 + 12 T + 13 T^{2} - 223 T^{3} + 326 T^{4} + 4537 T^{5} - 5551 T^{6} + 52236 T^{7} - 541651 T^{8} - 6074353 T^{9} + 33540303 T^{10} - 62852409 T^{11} - 3029364626 T^{12} - 62852409 p T^{13} + 33540303 p^{2} T^{14} - 6074353 p^{3} T^{15} - 541651 p^{4} T^{16} + 52236 p^{5} T^{17} - 5551 p^{6} T^{18} + 4537 p^{7} T^{19} + 326 p^{8} T^{20} - 223 p^{9} T^{21} + 13 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 13 T - 33 T^{2} + 1330 T^{3} - 5308 T^{4} - 45413 T^{5} + 538870 T^{6} - 1058624 T^{7} - 16353842 T^{8} + 128132797 T^{9} - 180178425 T^{10} - 2773787169 T^{11} + 25187757138 T^{12} - 2773787169 p T^{13} - 180178425 p^{2} T^{14} + 128132797 p^{3} T^{15} - 16353842 p^{4} T^{16} - 1058624 p^{5} T^{17} + 538870 p^{6} T^{18} - 45413 p^{7} T^{19} - 5308 p^{8} T^{20} + 1330 p^{9} T^{21} - 33 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 23 T + 183 T^{2} + 523 T^{3} + 2171 T^{4} + 47618 T^{5} + 413784 T^{6} + 2192484 T^{7} + 17878464 T^{8} + 165471518 T^{9} + 998460948 T^{10} + 4101545294 T^{11} + 18493818874 T^{12} + 4101545294 p T^{13} + 998460948 p^{2} T^{14} + 165471518 p^{3} T^{15} + 17878464 p^{4} T^{16} + 2192484 p^{5} T^{17} + 413784 p^{6} T^{18} + 47618 p^{7} T^{19} + 2171 p^{8} T^{20} + 523 p^{9} T^{21} + 183 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \)
43 \( ( 1 + 25 T + 443 T^{2} + 5515 T^{3} + 56835 T^{4} + 475510 T^{5} + 3415010 T^{6} + 475510 p T^{7} + 56835 p^{2} T^{8} + 5515 p^{3} T^{9} + 443 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 + T - 69 T^{2} + 239 T^{3} + 6477 T^{4} - 4432 T^{5} - 387564 T^{6} + 766988 T^{7} + 20113500 T^{8} - 34196416 T^{9} - 1070828686 T^{10} - 468011586 T^{11} + 48081764570 T^{12} - 468011586 p T^{13} - 1070828686 p^{2} T^{14} - 34196416 p^{3} T^{15} + 20113500 p^{4} T^{16} + 766988 p^{5} T^{17} - 387564 p^{6} T^{18} - 4432 p^{7} T^{19} + 6477 p^{8} T^{20} + 239 p^{9} T^{21} - 69 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 21 T + 113 T^{2} + 480 T^{3} - 3818 T^{4} - 66081 T^{5} + 764610 T^{6} - 1231332 T^{7} - 1322382 T^{8} - 297972771 T^{9} + 3254242025 T^{10} - 4778141607 T^{11} - 77922774362 T^{12} - 4778141607 p T^{13} + 3254242025 p^{2} T^{14} - 297972771 p^{3} T^{15} - 1322382 p^{4} T^{16} - 1231332 p^{5} T^{17} + 764610 p^{6} T^{18} - 66081 p^{7} T^{19} - 3818 p^{8} T^{20} + 480 p^{9} T^{21} + 113 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 9 T + 7 T^{2} - 529 T^{3} - 4049 T^{4} + 12814 T^{5} + 446296 T^{6} + 2350008 T^{7} - 10129056 T^{8} - 281765226 T^{9} - 1540417248 T^{10} + 6251689178 T^{11} + 130056196874 T^{12} + 6251689178 p T^{13} - 1540417248 p^{2} T^{14} - 281765226 p^{3} T^{15} - 10129056 p^{4} T^{16} + 2350008 p^{5} T^{17} + 446296 p^{6} T^{18} + 12814 p^{7} T^{19} - 4049 p^{8} T^{20} - 529 p^{9} T^{21} + 7 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 26 T + 211 T^{2} - 939 T^{3} - 25344 T^{4} - 117039 T^{5} + 38221 T^{6} - 4961444 T^{7} - 59685069 T^{8} + 405621811 T^{9} + 7923270111 T^{10} + 10091674181 T^{11} - 277516312934 T^{12} + 10091674181 p T^{13} + 7923270111 p^{2} T^{14} + 405621811 p^{3} T^{15} - 59685069 p^{4} T^{16} - 4961444 p^{5} T^{17} + 38221 p^{6} T^{18} - 117039 p^{7} T^{19} - 25344 p^{8} T^{20} - 939 p^{9} T^{21} + 211 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 - 37 T + 11 p T^{2} - 11665 T^{3} + 164147 T^{4} - 2062122 T^{5} + 23810210 T^{6} - 259823016 T^{7} + 2675533158 T^{8} - 25966581222 T^{9} + 239313744080 T^{10} - 2101508568736 T^{11} + 17602530247678 T^{12} - 2101508568736 p T^{13} + 239313744080 p^{2} T^{14} - 25966581222 p^{3} T^{15} + 2675533158 p^{4} T^{16} - 259823016 p^{5} T^{17} + 23810210 p^{6} T^{18} - 2062122 p^{7} T^{19} + 164147 p^{8} T^{20} - 11665 p^{9} T^{21} + 11 p^{11} T^{22} - 37 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 21 T + T^{2} - 19 p T^{3} + 10771 T^{4} + 67866 T^{5} - 264144 T^{6} + 20842296 T^{7} + 46063856 T^{8} - 1792845654 T^{9} + 7389179196 T^{10} + 96304225006 T^{11} - 413283203394 T^{12} + 96304225006 p T^{13} + 7389179196 p^{2} T^{14} - 1792845654 p^{3} T^{15} + 46063856 p^{4} T^{16} + 20842296 p^{5} T^{17} - 264144 p^{6} T^{18} + 67866 p^{7} T^{19} + 10771 p^{8} T^{20} - 19 p^{10} T^{21} + p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 18 T - 11 T^{2} + 1697 T^{3} - 6318 T^{4} - 96309 T^{5} + 2090249 T^{6} - 15143916 T^{7} - 53445245 T^{8} + 1395629633 T^{9} - 5747246199 T^{10} - 79066630723 T^{11} + 1328147128830 T^{12} - 79066630723 p T^{13} - 5747246199 p^{2} T^{14} + 1395629633 p^{3} T^{15} - 53445245 p^{4} T^{16} - 15143916 p^{5} T^{17} + 2090249 p^{6} T^{18} - 96309 p^{7} T^{19} - 6318 p^{8} T^{20} + 1697 p^{9} T^{21} - 11 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 24 T + 252 T^{2} - 2216 T^{3} + 27111 T^{4} - 297424 T^{5} + 2287436 T^{6} - 13926068 T^{7} + 93806084 T^{8} - 810763724 T^{9} + 6428940772 T^{10} - 17289470648 T^{11} - 104116056176 T^{12} - 17289470648 p T^{13} + 6428940772 p^{2} T^{14} - 810763724 p^{3} T^{15} + 93806084 p^{4} T^{16} - 13926068 p^{5} T^{17} + 2287436 p^{6} T^{18} - 297424 p^{7} T^{19} + 27111 p^{8} T^{20} - 2216 p^{9} T^{21} + 252 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 46 T + 11 p T^{2} - 11375 T^{3} + 136602 T^{4} - 1946041 T^{5} + 23993235 T^{6} - 220675522 T^{7} + 1922016543 T^{8} - 21103229401 T^{9} + 221198443925 T^{10} - 1760498099887 T^{11} + 13654984203298 T^{12} - 1760498099887 p T^{13} + 221198443925 p^{2} T^{14} - 21103229401 p^{3} T^{15} + 1922016543 p^{4} T^{16} - 220675522 p^{5} T^{17} + 23993235 p^{6} T^{18} - 1946041 p^{7} T^{19} + 136602 p^{8} T^{20} - 11375 p^{9} T^{21} + 11 p^{11} T^{22} - 46 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 2 T + 34 T^{2} + 98 T^{3} + 14261 T^{4} + 18542 T^{5} + 1756 p T^{6} + 5557948 T^{7} + 96398796 T^{8} + 548901022 T^{9} + 814939374 T^{10} + 68326526948 T^{11} + 536156406596 T^{12} + 68326526948 p T^{13} + 814939374 p^{2} T^{14} + 548901022 p^{3} T^{15} + 96398796 p^{4} T^{16} + 5557948 p^{5} T^{17} + 1756 p^{7} T^{18} + 18542 p^{7} T^{19} + 14261 p^{8} T^{20} + 98 p^{9} T^{21} + 34 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 + 7 T - 123 T^{2} - 1605 T^{3} + 657 T^{4} - 51348 T^{5} + 95680 T^{6} + 18416476 T^{7} + 105394968 T^{8} - 476041588 T^{9} + 10556842230 T^{10} - 65948957114 T^{11} - 2540582876162 T^{12} - 65948957114 p T^{13} + 10556842230 p^{2} T^{14} - 476041588 p^{3} T^{15} + 105394968 p^{4} T^{16} + 18416476 p^{5} T^{17} + 95680 p^{6} T^{18} - 51348 p^{7} T^{19} + 657 p^{8} T^{20} - 1605 p^{9} T^{21} - 123 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
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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

−3.74735449522805783996655114406, −3.66860350192539977779337753677, −3.51537242264570302436303846127, −3.45316622849073180784300125321, −3.42071658550057438273266403814, −3.39493309142032722150656963603, −3.24976169065892238608682713986, −3.20051692202431792934348186150, −3.08192308511892265998712084779, −2.90752997081673811320422693258, −2.75207569602248002702823967540, −2.62844458502954681135310785763, −2.14062049664358907269914871627, −1.99080681393859835604428320973, −1.94327408557532789611753685901, −1.93501076475691125092376705420, −1.92701430331381630856258792600, −1.72225087804392749625155779241, −1.70657374318819464551601521890, −1.46406032701561941422439387339, −1.27950840190121543595925163678, −0.826512997043015275928671102059, −0.820305264946501133612021259195, −0.39216927924568806853498011618, −0.35832441740836404894347443187, 0.35832441740836404894347443187, 0.39216927924568806853498011618, 0.820305264946501133612021259195, 0.826512997043015275928671102059, 1.27950840190121543595925163678, 1.46406032701561941422439387339, 1.70657374318819464551601521890, 1.72225087804392749625155779241, 1.92701430331381630856258792600, 1.93501076475691125092376705420, 1.94327408557532789611753685901, 1.99080681393859835604428320973, 2.14062049664358907269914871627, 2.62844458502954681135310785763, 2.75207569602248002702823967540, 2.90752997081673811320422693258, 3.08192308511892265998712084779, 3.20051692202431792934348186150, 3.24976169065892238608682713986, 3.39493309142032722150656963603, 3.42071658550057438273266403814, 3.45316622849073180784300125321, 3.51537242264570302436303846127, 3.66860350192539977779337753677, 3.74735449522805783996655114406

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

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