Properties

Degree 24
Conductor $ 2^{48} \cdot 3^{12} \cdot 167^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 12

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s + 4·5-s − 11·7-s + 78·9-s − 11-s + 8·13-s − 48·15-s + 3·17-s − 12·19-s + 132·21-s − 7·23-s − 13·25-s − 364·27-s + 5·29-s − 33·31-s + 12·33-s − 44·35-s + 8·37-s − 96·39-s − 6·41-s − 16·43-s + 312·45-s − 18·47-s + 31·49-s − 36·51-s + 20·53-s − 4·55-s + ⋯
L(s)  = 1  − 6.92·3-s + 1.78·5-s − 4.15·7-s + 26·9-s − 0.301·11-s + 2.21·13-s − 12.3·15-s + 0.727·17-s − 2.75·19-s + 28.8·21-s − 1.45·23-s − 2.59·25-s − 70.0·27-s + 0.928·29-s − 5.92·31-s + 2.08·33-s − 7.43·35-s + 1.31·37-s − 15.3·39-s − 0.937·41-s − 2.43·43-s + 46.5·45-s − 2.62·47-s + 31/7·49-s − 5.04·51-s + 2.74·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{48} \cdot 3^{12} \cdot 167^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  12
Selberg data  =  $(24,\ 2^{48} \cdot 3^{12} \cdot 167^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \(F_p\) is a polynomial of degree 24. If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad2 \( 1 \)
3 \( ( 1 + T )^{12} \)
167 \( ( 1 + T )^{12} \)
good5 \( 1 - 4 T + 29 T^{2} - 89 T^{3} + 409 T^{4} - 1058 T^{5} + 3913 T^{6} - 1821 p T^{7} + 29926 T^{8} - 64597 T^{9} + 38604 p T^{10} - 77299 p T^{11} + 1053908 T^{12} - 77299 p^{2} T^{13} + 38604 p^{3} T^{14} - 64597 p^{3} T^{15} + 29926 p^{4} T^{16} - 1821 p^{6} T^{17} + 3913 p^{6} T^{18} - 1058 p^{7} T^{19} + 409 p^{8} T^{20} - 89 p^{9} T^{21} + 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \)
7 \( 1 + 11 T + 90 T^{2} + 550 T^{3} + 2935 T^{4} + 1936 p T^{5} + 57133 T^{6} + 218439 T^{7} + 778294 T^{8} + 2566764 T^{9} + 7963941 T^{10} + 23067122 T^{11} + 63084108 T^{12} + 23067122 p T^{13} + 7963941 p^{2} T^{14} + 2566764 p^{3} T^{15} + 778294 p^{4} T^{16} + 218439 p^{5} T^{17} + 57133 p^{6} T^{18} + 1936 p^{8} T^{19} + 2935 p^{8} T^{20} + 550 p^{9} T^{21} + 90 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
11 \( 1 + T + 57 T^{2} - 9 T^{3} + 1303 T^{4} - 2130 T^{5} + 15693 T^{6} - 4980 p T^{7} + 142415 T^{8} - 536457 T^{9} + 1867410 T^{10} - 1451533 T^{11} + 25055730 T^{12} - 1451533 p T^{13} + 1867410 p^{2} T^{14} - 536457 p^{3} T^{15} + 142415 p^{4} T^{16} - 4980 p^{6} T^{17} + 15693 p^{6} T^{18} - 2130 p^{7} T^{19} + 1303 p^{8} T^{20} - 9 p^{9} T^{21} + 57 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \)
13 \( 1 - 8 T + 103 T^{2} - 48 p T^{3} + 4818 T^{4} - 23799 T^{5} + 141509 T^{6} - 600471 T^{7} + 3036479 T^{8} - 11489293 T^{9} + 51865524 T^{10} - 178767981 T^{11} + 735638636 T^{12} - 178767981 p T^{13} + 51865524 p^{2} T^{14} - 11489293 p^{3} T^{15} + 3036479 p^{4} T^{16} - 600471 p^{5} T^{17} + 141509 p^{6} T^{18} - 23799 p^{7} T^{19} + 4818 p^{8} T^{20} - 48 p^{10} T^{21} + 103 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
17 \( 1 - 3 T + 118 T^{2} - 387 T^{3} + 7251 T^{4} - 23602 T^{5} + 300910 T^{6} - 932520 T^{7} + 9256187 T^{8} - 26738797 T^{9} + 221263212 T^{10} - 584079475 T^{11} + 4205285970 T^{12} - 584079475 p T^{13} + 221263212 p^{2} T^{14} - 26738797 p^{3} T^{15} + 9256187 p^{4} T^{16} - 932520 p^{5} T^{17} + 300910 p^{6} T^{18} - 23602 p^{7} T^{19} + 7251 p^{8} T^{20} - 387 p^{9} T^{21} + 118 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \)
19 \( 1 + 12 T + 199 T^{2} + 1674 T^{3} + 16320 T^{4} + 107611 T^{5} + 41355 p T^{6} + 4319201 T^{7} + 26066607 T^{8} + 124863285 T^{9} + 660397808 T^{10} + 2851154001 T^{11} + 13691500992 T^{12} + 2851154001 p T^{13} + 660397808 p^{2} T^{14} + 124863285 p^{3} T^{15} + 26066607 p^{4} T^{16} + 4319201 p^{5} T^{17} + 41355 p^{7} T^{18} + 107611 p^{7} T^{19} + 16320 p^{8} T^{20} + 1674 p^{9} T^{21} + 199 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \)
23 \( 1 + 7 T + 167 T^{2} + 941 T^{3} + 13183 T^{4} + 2628 p T^{5} + 655771 T^{6} + 2465474 T^{7} + 23482855 T^{8} + 73464215 T^{9} + 666819294 T^{10} + 1819767099 T^{11} + 16262816722 T^{12} + 1819767099 p T^{13} + 666819294 p^{2} T^{14} + 73464215 p^{3} T^{15} + 23482855 p^{4} T^{16} + 2465474 p^{5} T^{17} + 655771 p^{6} T^{18} + 2628 p^{8} T^{19} + 13183 p^{8} T^{20} + 941 p^{9} T^{21} + 167 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \)
29 \( 1 - 5 T + 265 T^{2} - 939 T^{3} + 31695 T^{4} - 75812 T^{5} + 2339341 T^{6} - 3519780 T^{7} + 122806087 T^{8} - 110222539 T^{9} + 4959840978 T^{10} - 2882442317 T^{11} + 160066031378 T^{12} - 2882442317 p T^{13} + 4959840978 p^{2} T^{14} - 110222539 p^{3} T^{15} + 122806087 p^{4} T^{16} - 3519780 p^{5} T^{17} + 2339341 p^{6} T^{18} - 75812 p^{7} T^{19} + 31695 p^{8} T^{20} - 939 p^{9} T^{21} + 265 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \)
31 \( 1 + 33 T + 715 T^{2} + 11453 T^{3} + 151689 T^{4} + 1717753 T^{5} + 17166335 T^{6} + 153572178 T^{7} + 40185534 p T^{8} + 9228392127 T^{9} + 2025747070 p T^{10} + 393831808170 T^{11} + 2281105342288 T^{12} + 393831808170 p T^{13} + 2025747070 p^{3} T^{14} + 9228392127 p^{3} T^{15} + 40185534 p^{5} T^{16} + 153572178 p^{5} T^{17} + 17166335 p^{6} T^{18} + 1717753 p^{7} T^{19} + 151689 p^{8} T^{20} + 11453 p^{9} T^{21} + 715 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \)
37 \( 1 - 8 T + 199 T^{2} - 657 T^{3} + 15497 T^{4} - 11784 T^{5} + 1024693 T^{6} - 275281 T^{7} + 62717478 T^{8} + 2913275 T^{9} + 2925421190 T^{10} + 1550447139 T^{11} + 114197307556 T^{12} + 1550447139 p T^{13} + 2925421190 p^{2} T^{14} + 2913275 p^{3} T^{15} + 62717478 p^{4} T^{16} - 275281 p^{5} T^{17} + 1024693 p^{6} T^{18} - 11784 p^{7} T^{19} + 15497 p^{8} T^{20} - 657 p^{9} T^{21} + 199 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
41 \( 1 + 6 T + 6 p T^{2} + 1573 T^{3} + 32094 T^{4} + 205504 T^{5} + 2951176 T^{6} + 17984407 T^{7} + 208488291 T^{8} + 1180521459 T^{9} + 11680066058 T^{10} + 1483371931 p T^{11} + 529803935660 T^{12} + 1483371931 p^{2} T^{13} + 11680066058 p^{2} T^{14} + 1180521459 p^{3} T^{15} + 208488291 p^{4} T^{16} + 17984407 p^{5} T^{17} + 2951176 p^{6} T^{18} + 205504 p^{7} T^{19} + 32094 p^{8} T^{20} + 1573 p^{9} T^{21} + 6 p^{11} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \)
43 \( 1 + 16 T + 352 T^{2} + 4313 T^{3} + 57980 T^{4} + 594402 T^{5} + 6230176 T^{6} + 55553277 T^{7} + 491590559 T^{8} + 3876909777 T^{9} + 29940548104 T^{10} + 210459992337 T^{11} + 1441935098296 T^{12} + 210459992337 p T^{13} + 29940548104 p^{2} T^{14} + 3876909777 p^{3} T^{15} + 491590559 p^{4} T^{16} + 55553277 p^{5} T^{17} + 6230176 p^{6} T^{18} + 594402 p^{7} T^{19} + 57980 p^{8} T^{20} + 4313 p^{9} T^{21} + 352 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \)
47 \( 1 + 18 T + 425 T^{2} + 5660 T^{3} + 83071 T^{4} + 912185 T^{5} + 10346123 T^{6} + 97439810 T^{7} + 923239432 T^{8} + 7625094882 T^{9} + 62597370528 T^{10} + 457914714495 T^{11} + 3314420997080 T^{12} + 457914714495 p T^{13} + 62597370528 p^{2} T^{14} + 7625094882 p^{3} T^{15} + 923239432 p^{4} T^{16} + 97439810 p^{5} T^{17} + 10346123 p^{6} T^{18} + 912185 p^{7} T^{19} + 83071 p^{8} T^{20} + 5660 p^{9} T^{21} + 425 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \)
53 \( 1 - 20 T + 549 T^{2} - 144 p T^{3} + 122313 T^{4} - 1328735 T^{5} + 16067905 T^{6} - 147185648 T^{7} + 1496194206 T^{8} - 12145564678 T^{9} + 109080630288 T^{10} - 798675767107 T^{11} + 6425851329756 T^{12} - 798675767107 p T^{13} + 109080630288 p^{2} T^{14} - 12145564678 p^{3} T^{15} + 1496194206 p^{4} T^{16} - 147185648 p^{5} T^{17} + 16067905 p^{6} T^{18} - 1328735 p^{7} T^{19} + 122313 p^{8} T^{20} - 144 p^{10} T^{21} + 549 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \)
59 \( 1 + 4 T + 377 T^{2} + 1836 T^{3} + 73773 T^{4} + 392035 T^{5} + 9904875 T^{6} + 53653804 T^{7} + 1009471726 T^{8} + 5306097190 T^{9} + 81672478638 T^{10} + 400645649097 T^{11} + 5344564122780 T^{12} + 400645649097 p T^{13} + 81672478638 p^{2} T^{14} + 5306097190 p^{3} T^{15} + 1009471726 p^{4} T^{16} + 53653804 p^{5} T^{17} + 9904875 p^{6} T^{18} + 392035 p^{7} T^{19} + 73773 p^{8} T^{20} + 1836 p^{9} T^{21} + 377 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \)
61 \( 1 - 10 T + 445 T^{2} - 4156 T^{3} + 94520 T^{4} - 789057 T^{5} + 12579213 T^{6} - 91911701 T^{7} + 1186381407 T^{8} - 7600064489 T^{9} + 87750053054 T^{10} - 509722415835 T^{11} + 5618056854064 T^{12} - 509722415835 p T^{13} + 87750053054 p^{2} T^{14} - 7600064489 p^{3} T^{15} + 1186381407 p^{4} T^{16} - 91911701 p^{5} T^{17} + 12579213 p^{6} T^{18} - 789057 p^{7} T^{19} + 94520 p^{8} T^{20} - 4156 p^{9} T^{21} + 445 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \)
67 \( 1 + 9 T + 364 T^{2} + 2307 T^{3} + 64381 T^{4} + 335261 T^{5} + 8131027 T^{6} + 36716836 T^{7} + 807347350 T^{8} + 3186819725 T^{9} + 66922528995 T^{10} + 241334650304 T^{11} + 4818627301004 T^{12} + 241334650304 p T^{13} + 66922528995 p^{2} T^{14} + 3186819725 p^{3} T^{15} + 807347350 p^{4} T^{16} + 36716836 p^{5} T^{17} + 8131027 p^{6} T^{18} + 335261 p^{7} T^{19} + 64381 p^{8} T^{20} + 2307 p^{9} T^{21} + 364 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \)
71 \( 1 + 11 T + 396 T^{2} + 4588 T^{3} + 97063 T^{4} + 1016101 T^{5} + 16323044 T^{6} + 156812678 T^{7} + 2072326023 T^{8} + 18020382775 T^{9} + 206089443520 T^{10} + 1618074602797 T^{11} + 16319516903874 T^{12} + 1618074602797 p T^{13} + 206089443520 p^{2} T^{14} + 18020382775 p^{3} T^{15} + 2072326023 p^{4} T^{16} + 156812678 p^{5} T^{17} + 16323044 p^{6} T^{18} + 1016101 p^{7} T^{19} + 97063 p^{8} T^{20} + 4588 p^{9} T^{21} + 396 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \)
73 \( 1 - 22 T + 633 T^{2} - 9692 T^{3} + 162972 T^{4} - 1961473 T^{5} + 25060151 T^{6} - 256579429 T^{7} + 2800657223 T^{8} - 25744486769 T^{9} + 255423588848 T^{10} - 2167713986391 T^{11} + 20005755817240 T^{12} - 2167713986391 p T^{13} + 255423588848 p^{2} T^{14} - 25744486769 p^{3} T^{15} + 2800657223 p^{4} T^{16} - 256579429 p^{5} T^{17} + 25060151 p^{6} T^{18} - 1961473 p^{7} T^{19} + 162972 p^{8} T^{20} - 9692 p^{9} T^{21} + 633 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \)
79 \( 1 + 56 T + 1860 T^{2} + 44299 T^{3} + 844526 T^{4} + 13509480 T^{5} + 189081904 T^{6} + 2376082387 T^{7} + 27488453199 T^{8} + 296941458635 T^{9} + 3028562934604 T^{10} + 29215410638157 T^{11} + 267020165736708 T^{12} + 29215410638157 p T^{13} + 3028562934604 p^{2} T^{14} + 296941458635 p^{3} T^{15} + 27488453199 p^{4} T^{16} + 2376082387 p^{5} T^{17} + 189081904 p^{6} T^{18} + 13509480 p^{7} T^{19} + 844526 p^{8} T^{20} + 44299 p^{9} T^{21} + 1860 p^{10} T^{22} + 56 p^{11} T^{23} + p^{12} T^{24} \)
83 \( 1 + 26 T + 973 T^{2} + 17670 T^{3} + 376595 T^{4} + 5194783 T^{5} + 80296787 T^{6} + 880200858 T^{7} + 10936562682 T^{8} + 99486639658 T^{9} + 1085115323418 T^{10} + 8825068327103 T^{11} + 92536860445288 T^{12} + 8825068327103 p T^{13} + 1085115323418 p^{2} T^{14} + 99486639658 p^{3} T^{15} + 10936562682 p^{4} T^{16} + 880200858 p^{5} T^{17} + 80296787 p^{6} T^{18} + 5194783 p^{7} T^{19} + 376595 p^{8} T^{20} + 17670 p^{9} T^{21} + 973 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \)
89 \( 1 + 15 T + 264 T^{2} + 2377 T^{3} + 35623 T^{4} + 309093 T^{5} + 4329853 T^{6} + 33743512 T^{7} + 444605126 T^{8} + 3224059839 T^{9} + 42312991591 T^{10} + 313857383284 T^{11} + 4118090623756 T^{12} + 313857383284 p T^{13} + 42312991591 p^{2} T^{14} + 3224059839 p^{3} T^{15} + 444605126 p^{4} T^{16} + 33743512 p^{5} T^{17} + 4329853 p^{6} T^{18} + 309093 p^{7} T^{19} + 35623 p^{8} T^{20} + 2377 p^{9} T^{21} + 264 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \)
97 \( 1 - 8 T + 622 T^{2} - 5261 T^{3} + 199675 T^{4} - 1747832 T^{5} + 43601123 T^{6} - 383408405 T^{7} + 7204409176 T^{8} - 61549406631 T^{9} + 947657007183 T^{10} - 7599297626119 T^{11} + 101521370408824 T^{12} - 7599297626119 p T^{13} + 947657007183 p^{2} T^{14} - 61549406631 p^{3} T^{15} + 7204409176 p^{4} T^{16} - 383408405 p^{5} T^{17} + 43601123 p^{6} T^{18} - 1747832 p^{7} T^{19} + 199675 p^{8} T^{20} - 5261 p^{9} T^{21} + 622 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \)
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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

−2.81248096850084182819076346035, −2.47570042042564008254695925855, −2.44885449789271938426520359510, −2.29925402414912427514326742273, −2.28661055253267645327306074223, −2.23198996636842962619411684562, −2.17295243642728938082192621630, −2.12617946359101874835210196957, −2.11336000904444405780114555196, −2.00531465039884007892947710431, −1.99013232784379187151706134728, −1.92985697298117670519760484559, −1.87724672239813832801328460513, −1.53408474268995673574896956732, −1.37006065342879017111405049345, −1.34899557065709381453012415346, −1.31408101516674436445354031123, −1.30289685508463017918204614776, −1.28974588478810733114371552872, −1.28319423450769220425378093264, −1.08878409104575144431088540899, −1.05015736327502766854780313881, −1.04757541224054261584707964823, −0.970269029790897361051313473636, −0.75718475244733846112267937952, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.75718475244733846112267937952, 0.970269029790897361051313473636, 1.04757541224054261584707964823, 1.05015736327502766854780313881, 1.08878409104575144431088540899, 1.28319423450769220425378093264, 1.28974588478810733114371552872, 1.30289685508463017918204614776, 1.31408101516674436445354031123, 1.34899557065709381453012415346, 1.37006065342879017111405049345, 1.53408474268995673574896956732, 1.87724672239813832801328460513, 1.92985697298117670519760484559, 1.99013232784379187151706134728, 2.00531465039884007892947710431, 2.11336000904444405780114555196, 2.12617946359101874835210196957, 2.17295243642728938082192621630, 2.23198996636842962619411684562, 2.28661055253267645327306074223, 2.29925402414912427514326742273, 2.44885449789271938426520359510, 2.47570042042564008254695925855, 2.81248096850084182819076346035

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

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