Properties

Degree 16
Conductor $ 2^{48} \cdot 5^{8} $
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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{48} \cdot 5^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(4.117162765\)
\(L(\frac12)\)  \(\approx\)  \(4.117162765\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.36590640246297794449076848368, −4.26864425932921979508012515636, −3.91569259936469693538117750794, −3.73523338293967947581854227992, −3.60641523496034296960157088691, −3.58013914667909384033123606220, −3.58009702018744692993280109330, −3.04641866355727704039653927270, −2.99006423371546851500093409344, −2.88556525985170918819418590568, −2.63158289437345133413806386232, −2.53068781389446046473730807228, −2.39931365015476864714155616922, −2.07884099517618706255970374519, −2.02336723147824760647880790916, −1.89598081998202538454240165060, −1.62564952253153195137308559297, −1.46593884147138193218684711683, −1.29449181091983282571858998923, −1.06641592633823178225468432004, −0.810429624391299696056941661171, −0.70937336679334322433829747808, −0.47902289191581290682935811742, −0.43930909774794211219689381578, −0.10764378688552703728341938468, 0.10764378688552703728341938468, 0.43930909774794211219689381578, 0.47902289191581290682935811742, 0.70937336679334322433829747808, 0.810429624391299696056941661171, 1.06641592633823178225468432004, 1.29449181091983282571858998923, 1.46593884147138193218684711683, 1.62564952253153195137308559297, 1.89598081998202538454240165060, 2.02336723147824760647880790916, 2.07884099517618706255970374519, 2.39931365015476864714155616922, 2.53068781389446046473730807228, 2.63158289437345133413806386232, 2.88556525985170918819418590568, 2.99006423371546851500093409344, 3.04641866355727704039653927270, 3.58009702018744692993280109330, 3.58013914667909384033123606220, 3.60641523496034296960157088691, 3.73523338293967947581854227992, 3.91569259936469693538117750794, 4.26864425932921979508012515636, 4.36590640246297794449076848368

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

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