Properties

Degree 20
Conductor $ 2^{100} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 28·7-s − 108·9-s + 4·17-s + 276·23-s − 600·25-s + 368·31-s + 944·47-s − 1.37e3·49-s − 3.02e3·63-s + 3.46e3·71-s − 296·73-s + 4.41e3·79-s + 5.34e3·81-s + 88·89-s − 4·97-s + 6.95e3·103-s + 668·113-s + 112·119-s − 7.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 432·153-s + 157-s + ⋯
L(s)  = 1  + 1.51·7-s − 4·9-s + 0.0570·17-s + 2.50·23-s − 4.79·25-s + 2.13·31-s + 2.92·47-s − 3.99·49-s − 6.04·63-s + 5.79·71-s − 0.474·73-s + 6.28·79-s + 7.33·81-s + 0.104·89-s − 0.00418·97-s + 6.65·103-s + 0.556·113-s + 0.0862·119-s − 5.45·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.228·153-s + 0.000508·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{100}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{1024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((20,\ 2^{100} ,\ ( \ : [3/2]^{10} ),\ 1 )\)
\(L(2)\)  \(\approx\)  \(0.4506368658\)
\(L(\frac12)\)  \(\approx\)  \(0.4506368658\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 20. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 4 p^{3} T^{2} + 6317 T^{4} + 275312 T^{6} + 3302590 p T^{8} + 295366792 T^{10} + 3302590 p^{7} T^{12} + 275312 p^{12} T^{14} + 6317 p^{18} T^{16} + 4 p^{27} T^{18} + p^{30} T^{20} \)
5 \( 1 + 24 p^{2} T^{2} + 191789 T^{4} + 42513504 T^{6} + 7224666922 T^{8} + 994659832592 T^{10} + 7224666922 p^{6} T^{12} + 42513504 p^{12} T^{14} + 191789 p^{18} T^{16} + 24 p^{26} T^{18} + p^{30} T^{20} \)
7 \( ( 1 - 2 p T + 979 T^{2} - 5832 T^{3} + 372410 T^{4} - 662580 T^{5} + 372410 p^{3} T^{6} - 5832 p^{6} T^{7} + 979 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10} )^{2} \)
11 \( 1 + 60 p^{2} T^{2} + 24476093 T^{4} + 51189622576 T^{6} + 78543614693626 T^{8} + 105919424238644520 T^{10} + 78543614693626 p^{6} T^{12} + 51189622576 p^{12} T^{14} + 24476093 p^{18} T^{16} + 60 p^{26} T^{18} + p^{30} T^{20} \)
13 \( 1 + 72 p^{2} T^{2} + 76341149 T^{4} + 324341675296 T^{6} + 1028186938895082 T^{8} + 2540163312339385904 T^{10} + 1028186938895082 p^{6} T^{12} + 324341675296 p^{12} T^{14} + 76341149 p^{18} T^{16} + 72 p^{26} T^{18} + p^{30} T^{20} \)
17 \( ( 1 - 2 T + 12653 T^{2} - 102520 T^{3} + 98460610 T^{4} - 354493580 T^{5} + 98460610 p^{3} T^{6} - 102520 p^{6} T^{7} + 12653 p^{9} T^{8} - 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
19 \( 1 + 33948 T^{2} + 631492045 T^{4} + 8119820572464 T^{6} + 79259035180229178 T^{8} + \)\(60\!\cdots\!76\)\( T^{10} + 79259035180229178 p^{6} T^{12} + 8119820572464 p^{12} T^{14} + 631492045 p^{18} T^{16} + 33948 p^{24} T^{18} + p^{30} T^{20} \)
23 \( ( 1 - 6 p T + 47715 T^{2} - 4085400 T^{3} + 882303386 T^{4} - 56970028764 T^{5} + 882303386 p^{3} T^{6} - 4085400 p^{6} T^{7} + 47715 p^{9} T^{8} - 6 p^{13} T^{9} + p^{15} T^{10} )^{2} \)
29 \( 1 + 154168 T^{2} + 380663905 p T^{4} + 493373494152672 T^{6} + 16008852482124057194 T^{8} + \)\(42\!\cdots\!28\)\( T^{10} + 16008852482124057194 p^{6} T^{12} + 493373494152672 p^{12} T^{14} + 380663905 p^{19} T^{16} + 154168 p^{24} T^{18} + p^{30} T^{20} \)
31 \( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 7638677322 p^{3} T^{6} - 19809056 p^{6} T^{7} + 134043 p^{9} T^{8} - 184 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
37 \( 1 + 361064 T^{2} + 63572869101 T^{4} + 7160954906480288 T^{6} + \)\(56\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!92\)\( T^{10} + \)\(56\!\cdots\!06\)\( p^{6} T^{12} + 7160954906480288 p^{12} T^{14} + 63572869101 p^{18} T^{16} + 361064 p^{24} T^{18} + p^{30} T^{20} \)
41 \( ( 1 + 220509 T^{2} + 10715136 T^{3} + 24270968490 T^{4} + 1235215122432 T^{5} + 24270968490 p^{3} T^{6} + 10715136 p^{6} T^{7} + 220509 p^{9} T^{8} + p^{15} T^{10} )^{2} \)
43 \( 1 + 12836 p T^{2} + 145009588605 T^{4} + 24152295654927088 T^{6} + \)\(28\!\cdots\!30\)\( T^{8} + \)\(25\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!30\)\( p^{6} T^{12} + 24152295654927088 p^{12} T^{14} + 145009588605 p^{18} T^{16} + 12836 p^{25} T^{18} + p^{30} T^{20} \)
47 \( ( 1 - 472 T + 462219 T^{2} - 171516064 T^{3} + 90105579914 T^{4} - 25593405310224 T^{5} + 90105579914 p^{3} T^{6} - 171516064 p^{6} T^{7} + 462219 p^{9} T^{8} - 472 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
53 \( 1 + 525160 T^{2} + 112387404877 T^{4} + 12025532417207968 T^{6} + \)\(67\!\cdots\!30\)\( T^{8} + \)\(40\!\cdots\!00\)\( T^{10} + \)\(67\!\cdots\!30\)\( p^{6} T^{12} + 12025532417207968 p^{12} T^{14} + 112387404877 p^{18} T^{16} + 525160 p^{24} T^{18} + p^{30} T^{20} \)
59 \( 1 + 958284 T^{2} + 532967592733 T^{4} + 207330855222256112 T^{6} + \)\(61\!\cdots\!62\)\( T^{8} + \)\(40\!\cdots\!88\)\( p^{2} T^{10} + \)\(61\!\cdots\!62\)\( p^{6} T^{12} + 207330855222256112 p^{12} T^{14} + 532967592733 p^{18} T^{16} + 958284 p^{24} T^{18} + p^{30} T^{20} \)
61 \( 1 + 1146824 T^{2} + 8354214273 p T^{4} + 80080659050426912 T^{6} - \)\(14\!\cdots\!54\)\( T^{8} - \)\(80\!\cdots\!72\)\( T^{10} - \)\(14\!\cdots\!54\)\( p^{6} T^{12} + 80080659050426912 p^{12} T^{14} + 8354214273 p^{19} T^{16} + 1146824 p^{24} T^{18} + p^{30} T^{20} \)
67 \( 1 + 2130652 T^{2} + 2127009110445 T^{4} + 1334413095892987440 T^{6} + \)\(59\!\cdots\!58\)\( T^{8} + \)\(20\!\cdots\!52\)\( T^{10} + \)\(59\!\cdots\!58\)\( p^{6} T^{12} + 1334413095892987440 p^{12} T^{14} + 2127009110445 p^{18} T^{16} + 2130652 p^{24} T^{18} + p^{30} T^{20} \)
71 \( ( 1 - 1734 T + 2753587 T^{2} - 2611066280 T^{3} + 2272726706522 T^{4} - 1414434499369348 T^{5} + 2272726706522 p^{3} T^{6} - 2611066280 p^{6} T^{7} + 2753587 p^{9} T^{8} - 1734 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
73 \( ( 1 + 148 T + 1578093 T^{2} + 253463696 T^{3} + 1105903969594 T^{4} + 150341055768952 T^{5} + 1105903969594 p^{3} T^{6} + 253463696 p^{6} T^{7} + 1578093 p^{9} T^{8} + 148 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
79 \( ( 1 - 2208 T + 3816107 T^{2} - 54694528 p T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 4245684014154 p^{3} T^{6} - 54694528 p^{7} T^{7} + 3816107 p^{9} T^{8} - 2208 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
83 \( 1 + 2541516 T^{2} + 3317268679373 T^{4} + 33811498843796816 p T^{6} + \)\(18\!\cdots\!66\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(18\!\cdots\!66\)\( p^{6} T^{12} + 33811498843796816 p^{13} T^{14} + 3317268679373 p^{18} T^{16} + 2541516 p^{24} T^{18} + p^{30} T^{20} \)
89 \( ( 1 - 44 T + 1822557 T^{2} - 625587056 T^{3} + 1619388326906 T^{4} - 839474353817096 T^{5} + 1619388326906 p^{3} T^{6} - 625587056 p^{6} T^{7} + 1822557 p^{9} T^{8} - 44 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
97 \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 3231145430658 p^{3} T^{6} - 723000584 p^{6} T^{7} + 2565789 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.18853266757187638752687406080, −3.10233498839197194488813361005, −3.08736760774475990019420857726, −2.99304256882878182906510455231, −2.75412650814173250352747269890, −2.48697389390343085973778299815, −2.37911539515792419699052942342, −2.28883260798489212004260433735, −2.27974282296017960222069509311, −2.24329270540233609938183858733, −1.97656705322758935871948669161, −1.95025969989089373048976509320, −1.90377093689440825680717598967, −1.72875358780219454224521890928, −1.71493998677633296920465832346, −1.37068447099714985526463778394, −1.18487164950767454534058824419, −0.962157248111783288151881205879, −0.848821938234964779563975377470, −0.72215083035860880544702619973, −0.69109870301647899797521411219, −0.57239065623834175270607073818, −0.41400873766664121318632781215, −0.35776384759393791392904697280, −0.02596870603313918829568212620, 0.02596870603313918829568212620, 0.35776384759393791392904697280, 0.41400873766664121318632781215, 0.57239065623834175270607073818, 0.69109870301647899797521411219, 0.72215083035860880544702619973, 0.848821938234964779563975377470, 0.962157248111783288151881205879, 1.18487164950767454534058824419, 1.37068447099714985526463778394, 1.71493998677633296920465832346, 1.72875358780219454224521890928, 1.90377093689440825680717598967, 1.95025969989089373048976509320, 1.97656705322758935871948669161, 2.24329270540233609938183858733, 2.27974282296017960222069509311, 2.28883260798489212004260433735, 2.37911539515792419699052942342, 2.48697389390343085973778299815, 2.75412650814173250352747269890, 2.99304256882878182906510455231, 3.08736760774475990019420857726, 3.10233498839197194488813361005, 3.18853266757187638752687406080

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

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