Properties

Degree 24
Conductor $ 5^{24} \cdot 241^{12} $
Sign $1$
Motivic weight 1
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 − 3-s − 4-s + 3·6-s − 3·7-s + 11·8-s − 10·9-s + 22·11-s + 12-s + 5·13-s + 9·14-s − 6·16-s + 4·17-s + 30·18-s − 6·19-s + 3·21-s − 66·22-s − 32·23-s − 11·24-s − 15·26-s + 13·27-s + 3·28-s + 6·29-s + 8·31-s − 10·32-s − 22·33-s − 12·34-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s − 1/2·4-s + 1.22·6-s − 1.13·7-s + 3.88·8-s − 3.33·9-s + 6.63·11-s + 0.288·12-s + 1.38·13-s + 2.40·14-s − 3/2·16-s + 0.970·17-s + 7.07·18-s − 1.37·19-s + 0.654·21-s − 14.0·22-s − 6.67·23-s − 2.24·24-s − 2.94·26-s + 2.50·27-s + 0.566·28-s + 1.11·29-s + 1.43·31-s − 1.76·32-s − 3.82·33-s − 2.05·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{24} \cdot 241^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{24} \cdot 241^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(5^{24} \cdot 241^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6025} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 5^{24} \cdot 241^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $0.4026920364$
$L(\frac12)$  $\approx$  $0.4026920364$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;241\}$, \(F_p\) is a polynomial of degree 24. If $p \in \{5,\;241\}$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad5 \( 1 \)
241 \( ( 1 - T )^{12} \)
good2 \( 1 + 3 T + 5 p T^{2} + 11 p T^{3} + 49 T^{4} + 87 T^{5} + 157 T^{6} + 123 p T^{7} + 389 T^{8} + 139 p^{2} T^{9} + 839 T^{10} + 579 p T^{11} + 1699 T^{12} + 579 p^{2} T^{13} + 839 p^{2} T^{14} + 139 p^{5} T^{15} + 389 p^{4} T^{16} + 123 p^{6} T^{17} + 157 p^{6} T^{18} + 87 p^{7} T^{19} + 49 p^{8} T^{20} + 11 p^{10} T^{21} + 5 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
3 \( 1 + T + 11 T^{2} + 8 T^{3} + 68 T^{4} + 10 p T^{5} + 101 p T^{6} + 40 T^{7} + 1099 T^{8} - 65 p T^{9} + 3502 T^{10} - 1444 T^{11} + 10528 T^{12} - 1444 p T^{13} + 3502 p^{2} T^{14} - 65 p^{4} T^{15} + 1099 p^{4} T^{16} + 40 p^{5} T^{17} + 101 p^{7} T^{18} + 10 p^{8} T^{19} + 68 p^{8} T^{20} + 8 p^{9} T^{21} + 11 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 + 3 T + 51 T^{2} + 135 T^{3} + 167 p T^{4} + 59 p^{2} T^{5} + 16678 T^{6} + 41432 T^{7} + 25259 p T^{8} + 460010 T^{9} + 1555215 T^{10} + 4088130 T^{11} + 11751926 T^{12} + 4088130 p T^{13} + 1555215 p^{2} T^{14} + 460010 p^{3} T^{15} + 25259 p^{5} T^{16} + 41432 p^{5} T^{17} + 16678 p^{6} T^{18} + 59 p^{9} T^{19} + 167 p^{9} T^{20} + 135 p^{9} T^{21} + 51 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 - 2 p T + 309 T^{2} - 3215 T^{3} + 27241 T^{4} - 195612 T^{5} + 1224926 T^{6} - 6803622 T^{7} + 33951641 T^{8} - 153501494 T^{9} + 632754137 T^{10} - 2387213463 T^{11} + 8263362466 T^{12} - 2387213463 p T^{13} + 632754137 p^{2} T^{14} - 153501494 p^{3} T^{15} + 33951641 p^{4} T^{16} - 6803622 p^{5} T^{17} + 1224926 p^{6} T^{18} - 195612 p^{7} T^{19} + 27241 p^{8} T^{20} - 3215 p^{9} T^{21} + 309 p^{10} T^{22} - 2 p^{12} T^{23} + p^{12} T^{24} \)
13 \( 1 - 5 T + 94 T^{2} - 419 T^{3} + 4519 T^{4} - 18313 T^{5} + 144981 T^{6} - 535786 T^{7} + 3431095 T^{8} - 11548019 T^{9} + 62910413 T^{10} - 191545534 T^{11} + 914969618 T^{12} - 191545534 p T^{13} + 62910413 p^{2} T^{14} - 11548019 p^{3} T^{15} + 3431095 p^{4} T^{16} - 535786 p^{5} T^{17} + 144981 p^{6} T^{18} - 18313 p^{7} T^{19} + 4519 p^{8} T^{20} - 419 p^{9} T^{21} + 94 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 4 T + 107 T^{2} - 378 T^{3} + 5581 T^{4} - 16942 T^{5} + 191746 T^{6} - 492741 T^{7} + 4974231 T^{8} - 10970245 T^{9} + 105662523 T^{10} - 208156762 T^{11} + 1924983462 T^{12} - 208156762 p T^{13} + 105662523 p^{2} T^{14} - 10970245 p^{3} T^{15} + 4974231 p^{4} T^{16} - 492741 p^{5} T^{17} + 191746 p^{6} T^{18} - 16942 p^{7} T^{19} + 5581 p^{8} T^{20} - 378 p^{9} T^{21} + 107 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 6 T + 142 T^{2} + 730 T^{3} + 10024 T^{4} + 2443 p T^{5} + 468739 T^{6} + 1979928 T^{7} + 16137130 T^{8} + 62321901 T^{9} + 430966450 T^{10} + 1510794001 T^{11} + 9148767508 T^{12} + 1510794001 p T^{13} + 430966450 p^{2} T^{14} + 62321901 p^{3} T^{15} + 16137130 p^{4} T^{16} + 1979928 p^{5} T^{17} + 468739 p^{6} T^{18} + 2443 p^{8} T^{19} + 10024 p^{8} T^{20} + 730 p^{9} T^{21} + 142 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 32 T + 580 T^{2} + 7469 T^{3} + 76528 T^{4} + 663240 T^{5} + 5077858 T^{6} + 35272611 T^{7} + 225993748 T^{8} + 1346655168 T^{9} + 7505092384 T^{10} + 39288214509 T^{11} + 193919253422 T^{12} + 39288214509 p T^{13} + 7505092384 p^{2} T^{14} + 1346655168 p^{3} T^{15} + 225993748 p^{4} T^{16} + 35272611 p^{5} T^{17} + 5077858 p^{6} T^{18} + 663240 p^{7} T^{19} + 76528 p^{8} T^{20} + 7469 p^{9} T^{21} + 580 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 6 T + 135 T^{2} - 539 T^{3} + 8952 T^{4} - 35377 T^{5} + 16518 p T^{6} - 1973598 T^{7} + 19968664 T^{8} - 81475880 T^{9} + 699787907 T^{10} - 2930882253 T^{11} + 22109454828 T^{12} - 2930882253 p T^{13} + 699787907 p^{2} T^{14} - 81475880 p^{3} T^{15} + 19968664 p^{4} T^{16} - 1973598 p^{5} T^{17} + 16518 p^{7} T^{18} - 35377 p^{7} T^{19} + 8952 p^{8} T^{20} - 539 p^{9} T^{21} + 135 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 - 8 T + 110 T^{2} - 561 T^{3} + 5136 T^{4} - 26697 T^{5} + 222132 T^{6} - 1395873 T^{7} + 10324443 T^{8} - 60423563 T^{9} + 375353534 T^{10} - 1867603862 T^{11} + 11382213016 T^{12} - 1867603862 p T^{13} + 375353534 p^{2} T^{14} - 60423563 p^{3} T^{15} + 10324443 p^{4} T^{16} - 1395873 p^{5} T^{17} + 222132 p^{6} T^{18} - 26697 p^{7} T^{19} + 5136 p^{8} T^{20} - 561 p^{9} T^{21} + 110 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 8 T + 285 T^{2} - 2328 T^{3} + 41990 T^{4} - 329585 T^{5} + 4122834 T^{6} - 29895592 T^{7} + 295299907 T^{8} - 1931900130 T^{9} + 16051467481 T^{10} - 93483683185 T^{11} + 673998628252 T^{12} - 93483683185 p T^{13} + 16051467481 p^{2} T^{14} - 1931900130 p^{3} T^{15} + 295299907 p^{4} T^{16} - 29895592 p^{5} T^{17} + 4122834 p^{6} T^{18} - 329585 p^{7} T^{19} + 41990 p^{8} T^{20} - 2328 p^{9} T^{21} + 285 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + T + 230 T^{2} - 257 T^{3} + 24637 T^{4} - 76060 T^{5} + 1796100 T^{6} - 7778661 T^{7} + 108454823 T^{8} - 482085656 T^{9} + 5703268056 T^{10} - 22805439650 T^{11} + 255177914992 T^{12} - 22805439650 p T^{13} + 5703268056 p^{2} T^{14} - 482085656 p^{3} T^{15} + 108454823 p^{4} T^{16} - 7778661 p^{5} T^{17} + 1796100 p^{6} T^{18} - 76060 p^{7} T^{19} + 24637 p^{8} T^{20} - 257 p^{9} T^{21} + 230 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 - 2 T + 279 T^{2} - 920 T^{3} + 38932 T^{4} - 175056 T^{5} + 3671987 T^{6} - 19664643 T^{7} + 264316714 T^{8} - 1510203737 T^{9} + 15303609823 T^{10} - 85368924579 T^{11} + 725763544528 T^{12} - 85368924579 p T^{13} + 15303609823 p^{2} T^{14} - 1510203737 p^{3} T^{15} + 264316714 p^{4} T^{16} - 19664643 p^{5} T^{17} + 3671987 p^{6} T^{18} - 175056 p^{7} T^{19} + 38932 p^{8} T^{20} - 920 p^{9} T^{21} + 279 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 34 T + 896 T^{2} + 16727 T^{3} + 267326 T^{4} + 3590905 T^{5} + 43102036 T^{6} + 460340779 T^{7} + 4494844159 T^{8} + 40090808867 T^{9} + 331074847308 T^{10} + 2529205380764 T^{11} + 17993329433412 T^{12} + 2529205380764 p T^{13} + 331074847308 p^{2} T^{14} + 40090808867 p^{3} T^{15} + 4494844159 p^{4} T^{16} + 460340779 p^{5} T^{17} + 43102036 p^{6} T^{18} + 3590905 p^{7} T^{19} + 267326 p^{8} T^{20} + 16727 p^{9} T^{21} + 896 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 + 5 T + 441 T^{2} + 1896 T^{3} + 94214 T^{4} + 351860 T^{5} + 12993530 T^{6} + 42520621 T^{7} + 1294765158 T^{8} + 3740747497 T^{9} + 98490903865 T^{10} + 252621044096 T^{11} + 5872652829352 T^{12} + 252621044096 p T^{13} + 98490903865 p^{2} T^{14} + 3740747497 p^{3} T^{15} + 1294765158 p^{4} T^{16} + 42520621 p^{5} T^{17} + 12993530 p^{6} T^{18} + 351860 p^{7} T^{19} + 94214 p^{8} T^{20} + 1896 p^{9} T^{21} + 441 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 - 26 T + 730 T^{2} - 11536 T^{3} + 185282 T^{4} - 2076094 T^{5} + 23928077 T^{6} - 201664335 T^{7} + 1842506089 T^{8} - 12100923663 T^{9} + 98573053649 T^{10} - 562897464460 T^{11} + 5101887162824 T^{12} - 562897464460 p T^{13} + 98573053649 p^{2} T^{14} - 12100923663 p^{3} T^{15} + 1842506089 p^{4} T^{16} - 201664335 p^{5} T^{17} + 23928077 p^{6} T^{18} - 2076094 p^{7} T^{19} + 185282 p^{8} T^{20} - 11536 p^{9} T^{21} + 730 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 + 26 T + 752 T^{2} + 13491 T^{3} + 235281 T^{4} + 3266857 T^{5} + 43103756 T^{6} + 492402970 T^{7} + 5338943579 T^{8} + 52076341808 T^{9} + 483789691118 T^{10} + 4119791757429 T^{11} + 33509307831364 T^{12} + 4119791757429 p T^{13} + 483789691118 p^{2} T^{14} + 52076341808 p^{3} T^{15} + 5338943579 p^{4} T^{16} + 492402970 p^{5} T^{17} + 43103756 p^{6} T^{18} + 3266857 p^{7} T^{19} + 235281 p^{8} T^{20} + 13491 p^{9} T^{21} + 752 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 6 T + 375 T^{2} + 1475 T^{3} + 71151 T^{4} + 178925 T^{5} + 9349095 T^{6} + 14634395 T^{7} + 969096135 T^{8} + 986114059 T^{9} + 83407729450 T^{10} + 63325311176 T^{11} + 6068485120114 T^{12} + 63325311176 p T^{13} + 83407729450 p^{2} T^{14} + 986114059 p^{3} T^{15} + 969096135 p^{4} T^{16} + 14634395 p^{5} T^{17} + 9349095 p^{6} T^{18} + 178925 p^{7} T^{19} + 71151 p^{8} T^{20} + 1475 p^{9} T^{21} + 375 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 - 94 T + 4589 T^{2} - 153184 T^{3} + 3904973 T^{4} - 80682013 T^{5} + 1402386991 T^{6} - 21038436416 T^{7} + 277565120451 T^{8} - 3266755261035 T^{9} + 34672343703145 T^{10} - 334475660641729 T^{11} + 2946747864969272 T^{12} - 334475660641729 p T^{13} + 34672343703145 p^{2} T^{14} - 3266755261035 p^{3} T^{15} + 277565120451 p^{4} T^{16} - 21038436416 p^{5} T^{17} + 1402386991 p^{6} T^{18} - 80682013 p^{7} T^{19} + 3904973 p^{8} T^{20} - 153184 p^{9} T^{21} + 4589 p^{10} T^{22} - 94 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 22 T + 668 T^{2} - 9806 T^{3} + 171777 T^{4} - 1817947 T^{5} + 22903099 T^{6} - 169880788 T^{7} + 1651008275 T^{8} - 6379023100 T^{9} + 53269643441 T^{10} + 183530814455 T^{11} + 710939601926 T^{12} + 183530814455 p T^{13} + 53269643441 p^{2} T^{14} - 6379023100 p^{3} T^{15} + 1651008275 p^{4} T^{16} - 169880788 p^{5} T^{17} + 22903099 p^{6} T^{18} - 1817947 p^{7} T^{19} + 171777 p^{8} T^{20} - 9806 p^{9} T^{21} + 668 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 - 9 T + 367 T^{2} - 2038 T^{3} + 62223 T^{4} - 141043 T^{5} + 6475151 T^{6} + 14731329 T^{7} + 423885699 T^{8} + 4946425456 T^{9} + 15523331202 T^{10} + 8089192135 p T^{11} + 470674369434 T^{12} + 8089192135 p^{2} T^{13} + 15523331202 p^{2} T^{14} + 4946425456 p^{3} T^{15} + 423885699 p^{4} T^{16} + 14731329 p^{5} T^{17} + 6475151 p^{6} T^{18} - 141043 p^{7} T^{19} + 62223 p^{8} T^{20} - 2038 p^{9} T^{21} + 367 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 - 8 T + 448 T^{2} - 2918 T^{3} + 100176 T^{4} - 424192 T^{5} + 13340059 T^{6} - 13375319 T^{7} + 1118067097 T^{8} + 5037020331 T^{9} + 59946946589 T^{10} + 927626542684 T^{11} + 3267800357708 T^{12} + 927626542684 p T^{13} + 59946946589 p^{2} T^{14} + 5037020331 p^{3} T^{15} + 1118067097 p^{4} T^{16} - 13375319 p^{5} T^{17} + 13340059 p^{6} T^{18} - 424192 p^{7} T^{19} + 100176 p^{8} T^{20} - 2918 p^{9} T^{21} + 448 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 3 T + 589 T^{2} + 1274 T^{3} + 175478 T^{4} + 280583 T^{5} + 35372418 T^{6} + 43704857 T^{7} + 5387584795 T^{8} + 5415512026 T^{9} + 651937772393 T^{10} + 561070696873 T^{11} + 64130514360812 T^{12} + 561070696873 p T^{13} + 651937772393 p^{2} T^{14} + 5415512026 p^{3} T^{15} + 5387584795 p^{4} T^{16} + 43704857 p^{5} T^{17} + 35372418 p^{6} T^{18} + 280583 p^{7} T^{19} + 175478 p^{8} T^{20} + 1274 p^{9} T^{21} + 589 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 29 T + 1249 T^{2} - 25366 T^{3} + 632678 T^{4} - 10020564 T^{5} + 185253972 T^{6} - 2425870975 T^{7} + 36546622034 T^{8} - 409964738669 T^{9} + 5276102462169 T^{10} - 51721589331614 T^{11} + 581746423923696 T^{12} - 51721589331614 p T^{13} + 5276102462169 p^{2} T^{14} - 409964738669 p^{3} T^{15} + 36546622034 p^{4} T^{16} - 2425870975 p^{5} T^{17} + 185253972 p^{6} T^{18} - 10020564 p^{7} T^{19} + 632678 p^{8} T^{20} - 25366 p^{9} T^{21} + 1249 p^{10} T^{22} - 29 p^{11} T^{23} + p^{12} T^{24} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.36662257398082114027121702059, −2.18857653605444168935949099546, −2.13019004690286069492745712142, −2.09547666587401544829821476218, −1.95611075712301800360198119634, −1.94761472284328676870295021048, −1.90783121947290207155352741782, −1.60136278667572095260678791810, −1.56648190417741679576757448773, −1.53050716002621438157860898574, −1.51644407978302126561511551911, −1.45869755532248128003965703726, −1.43589432747494202326951144168, −1.36163713423395386192194053971, −1.12282353191083303328075216364, −0.963880784943121677122008610201, −0.898275218291623688770780709962, −0.71756481672008946420241088002, −0.64155995647407857965767449583, −0.51385270651360754836525678170, −0.50076888313895986608905428779, −0.45270432634263638809535743554, −0.38475321813359077540418061694, −0.30394124144063704065703729827, −0.079357232701144311765082403836, 0.079357232701144311765082403836, 0.30394124144063704065703729827, 0.38475321813359077540418061694, 0.45270432634263638809535743554, 0.50076888313895986608905428779, 0.51385270651360754836525678170, 0.64155995647407857965767449583, 0.71756481672008946420241088002, 0.898275218291623688770780709962, 0.963880784943121677122008610201, 1.12282353191083303328075216364, 1.36163713423395386192194053971, 1.43589432747494202326951144168, 1.45869755532248128003965703726, 1.51644407978302126561511551911, 1.53050716002621438157860898574, 1.56648190417741679576757448773, 1.60136278667572095260678791810, 1.90783121947290207155352741782, 1.94761472284328676870295021048, 1.95611075712301800360198119634, 2.09547666587401544829821476218, 2.13019004690286069492745712142, 2.18857653605444168935949099546, 2.36662257398082114027121702059

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

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