Properties

Degree $16$
Conductor $1.678\times 10^{15}$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s + 472·9-s + 736·11-s − 1.37e3·19-s − 1.03e3·25-s + 5.87e3·29-s − 4.22e3·31-s + 2.36e4·41-s + 3.77e3·45-s + 4.47e4·49-s + 5.88e3·55-s − 9.16e4·59-s + 1.23e5·61-s + 1.25e5·71-s − 4.32e4·79-s + 1.53e5·81-s − 4.19e4·89-s − 1.10e4·95-s + 3.47e5·99-s + 6.72e5·101-s − 5.39e5·109-s + 5.38e4·121-s − 1.30e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.143·5-s + 1.94·9-s + 1.83·11-s − 0.874·19-s − 0.331·25-s + 1.29·29-s − 0.789·31-s + 2.19·41-s + 0.277·45-s + 2.66·49-s + 0.262·55-s − 3.42·59-s + 4.26·61-s + 2.95·71-s − 0.779·79-s + 2.59·81-s − 0.560·89-s − 0.125·95-s + 3.56·99-s + 6.55·101-s − 4.35·109-s + 0.334·121-s − 0.744·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{80} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(16.6415\)
\(L(\frac12)\) \(\approx\) \(16.6415\)
\(L(\frac{7}{2})\) 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 \( 1 - 8 T + 44 p^{2} T^{2} + 904 p^{3} T^{3} - 3506 p^{3} T^{4} + 904 p^{8} T^{5} + 44 p^{12} T^{6} - 8 p^{15} T^{7} + p^{20} T^{8} \)
good3 \( 1 - 472 T^{2} + 69724 T^{4} - 2235496 p^{2} T^{6} + 104192518 p^{4} T^{8} - 2235496 p^{12} T^{10} + 69724 p^{20} T^{12} - 472 p^{30} T^{14} + p^{40} T^{16} \)
7 \( 1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} - 37445733732616 p^{10} T^{10} + 1493128636 p^{20} T^{12} - 44728 p^{30} T^{14} + p^{40} T^{16} \)
11 \( ( 1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 51158896 p^{5} T^{5} + 176204 p^{10} T^{6} - 368 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
13 \( 1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - 437549632721743384 p^{10} T^{10} + 1074189263356 p^{20} T^{12} - 1578472 p^{30} T^{14} + p^{40} T^{16} \)
17 \( 1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} + \)\(45\!\cdots\!30\)\( T^{8} - 31297293718759478968 p^{10} T^{10} + 17547242668444 p^{20} T^{12} - 6260872 p^{30} T^{14} + p^{40} T^{16} \)
19 \( ( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 6058136368 p^{5} T^{5} + 5308396 p^{10} T^{6} + 688 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
23 \( 1 - 34675896 T^{2} + 569897415616828 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!82\)\( T^{8} - \)\(59\!\cdots\!20\)\( p^{10} T^{10} + 569897415616828 p^{20} T^{12} - 34675896 p^{30} T^{14} + p^{40} T^{16} \)
29 \( ( 1 - 2936 T + 58625996 T^{2} - 77951973928 T^{3} + 1466411094282230 T^{4} - 77951973928 p^{5} T^{5} + 58625996 p^{10} T^{6} - 2936 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
31 \( ( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 163080265536 p^{5} T^{5} + 80187004 p^{10} T^{6} + 2112 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
37 \( 1 - 251774632 T^{2} + 38631208311838780 T^{4} - \)\(41\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!34\)\( T^{8} - \)\(41\!\cdots\!52\)\( p^{10} T^{10} + 38631208311838780 p^{20} T^{12} - 251774632 p^{30} T^{14} + p^{40} T^{16} \)
41 \( ( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - 2943020124776 p^{5} T^{5} + 337909340 p^{10} T^{6} - 11800 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
43 \( 1 - 283211672 T^{2} + 48021567531024796 T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(54\!\cdots\!84\)\( p^{10} T^{10} + 48021567531024796 p^{20} T^{12} - 283211672 p^{30} T^{14} + p^{40} T^{16} \)
47 \( 1 - 963352312 T^{2} + 473832864723586300 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(15\!\cdots\!72\)\( p^{10} T^{10} + 473832864723586300 p^{20} T^{12} - 963352312 p^{30} T^{14} + p^{40} T^{16} \)
53 \( 1 - 1183385640 T^{2} + 874785099161623996 T^{4} - \)\(49\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(49\!\cdots\!80\)\( p^{10} T^{10} + 874785099161623996 p^{20} T^{12} - 1183385640 p^{30} T^{14} + p^{40} T^{16} \)
59 \( ( 1 + 45840 T + 3064286732 T^{2} + 94721285480976 T^{3} + 3348109683185502486 T^{4} + 94721285480976 p^{5} T^{5} + 3064286732 p^{10} T^{6} + 45840 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
61 \( ( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - 145287706763384 p^{5} T^{5} + 3903014764 p^{10} T^{6} - 61928 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
67 \( 1 - 9281919064 T^{2} + 39492482666681482588 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(10\!\cdots\!40\)\( p^{10} T^{10} + 39492482666681482588 p^{20} T^{12} - 9281919064 p^{30} T^{14} + p^{40} T^{16} \)
71 \( ( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} - 184024084124896 p^{5} T^{5} + 3398787356 p^{10} T^{6} - 62816 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
73 \( 1 - 9140679496 T^{2} + 46078306824990298588 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!02\)\( T^{8} - \)\(15\!\cdots\!60\)\( p^{10} T^{10} + 46078306824990298588 p^{20} T^{12} - 9140679496 p^{30} T^{14} + p^{40} T^{16} \)
79 \( ( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} + 332616618908288 p^{5} T^{5} + 7152876604 p^{10} T^{6} + 21632 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
83 \( 1 - 81444552 p T^{2} + 40943120759345365468 T^{4} - \)\(86\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!42\)\( T^{8} - \)\(86\!\cdots\!80\)\( p^{10} T^{10} + 40943120759345365468 p^{20} T^{12} - 81444552 p^{31} T^{14} + p^{40} T^{16} \)
89 \( ( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} + 497915996461992 p^{5} T^{5} + 16118164796 p^{10} T^{6} + 20952 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
97 \( 1 - 45263915272 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(14\!\cdots\!12\)\( p^{10} T^{10} + \)\(10\!\cdots\!40\)\( p^{20} T^{12} - 45263915272 p^{30} T^{14} + p^{40} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.70826850914543106217638717547, −5.42408719410674335134936437820, −5.34581042469028936698343582556, −5.34370140244901694473813929834, −4.78301319564271191630490035879, −4.57640843472720311246107917596, −4.55491566531707498269448921742, −4.32848171231335642919762942709, −4.20472287989945214391073053561, −3.94363954282492289054278688498, −3.84390793311885044065424965376, −3.70248155472038926440876750915, −3.39038826658155224344875657145, −3.09247284878212337570664882686, −2.95768738128760157675755817947, −2.40714591501446921536240570874, −2.39190260574833468501553529539, −2.02433941547405145491299286035, −1.82504486214790092607405989806, −1.81151780013200816661634208893, −1.13852016970987412421399429156, −1.03325997884725435930193708824, −0.909576438210600830714838108878, −0.63267455149793261088140427788, −0.32147471482301591902389277579, 0.32147471482301591902389277579, 0.63267455149793261088140427788, 0.909576438210600830714838108878, 1.03325997884725435930193708824, 1.13852016970987412421399429156, 1.81151780013200816661634208893, 1.82504486214790092607405989806, 2.02433941547405145491299286035, 2.39190260574833468501553529539, 2.40714591501446921536240570874, 2.95768738128760157675755817947, 3.09247284878212337570664882686, 3.39038826658155224344875657145, 3.70248155472038926440876750915, 3.84390793311885044065424965376, 3.94363954282492289054278688498, 4.20472287989945214391073053561, 4.32848171231335642919762942709, 4.55491566531707498269448921742, 4.57640843472720311246107917596, 4.78301319564271191630490035879, 5.34370140244901694473813929834, 5.34581042469028936698343582556, 5.42408719410674335134936437820, 5.70826850914543106217638717547

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

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