Properties

Degree 16
Conductor $ 2^{40} \cdot 7^{8} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·9-s + 32·13-s − 16·17-s − 48·25-s − 80·29-s − 176·37-s + 144·41-s − 28·49-s + 48·53-s − 192·61-s + 272·73-s + 164·81-s − 80·89-s + 528·97-s − 128·101-s + 208·109-s − 160·113-s + 512·117-s + 296·121-s + 448·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 256·153-s + ⋯
L(s)  = 1  + 16/9·9-s + 2.46·13-s − 0.941·17-s − 1.91·25-s − 2.75·29-s − 4.75·37-s + 3.51·41-s − 4/7·49-s + 0.905·53-s − 3.14·61-s + 3.72·73-s + 2.02·81-s − 0.898·89-s + 5.44·97-s − 1.26·101-s + 1.90·109-s − 1.41·113-s + 4.37·117-s + 2.44·121-s + 3.58·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 1.67·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{40} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.98554\)
\(L(\frac12)\)  \(\approx\)  \(2.98554\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 + p T^{2} )^{4} \)
good3 \( 1 - 16 T^{2} + 92 T^{4} + 272 T^{6} - 8570 T^{8} + 272 p^{4} T^{10} + 92 p^{8} T^{12} - 16 p^{12} T^{14} + p^{16} T^{16} \)
5 \( ( 1 + 24 T^{2} - 224 T^{3} + 78 T^{4} - 224 p^{2} T^{5} + 24 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( 1 - 296 T^{2} + 72860 T^{4} - 12907416 T^{6} + 1704788486 T^{8} - 12907416 p^{4} T^{10} + 72860 p^{8} T^{12} - 296 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 - 16 T + 696 T^{2} - 7536 T^{3} + 176398 T^{4} - 7536 p^{2} T^{5} + 696 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 1416 p^{2} T^{5} + 428 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
19 \( 1 - 1936 T^{2} + 1795420 T^{4} - 1065665392 T^{6} + 449725089670 T^{8} - 1065665392 p^{4} T^{10} + 1795420 p^{8} T^{12} - 1936 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 1917103032 p^{4} T^{10} + 2366876 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 40 T + 3660 T^{2} + 100632 T^{3} + 4741126 T^{4} + 100632 p^{2} T^{5} + 3660 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
31 \( 1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5382616134 p^{2} T^{8} - 5905968152 p^{4} T^{10} + 5953500 p^{8} T^{12} - 3624 p^{12} T^{14} + p^{16} T^{16} \)
37 \( ( 1 + 88 T + 3692 T^{2} + 66024 T^{3} + 510982 T^{4} + 66024 p^{2} T^{5} + 3692 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 72 T + 4364 T^{2} - 180408 T^{3} + 8880422 T^{4} - 180408 p^{2} T^{5} + 4364 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
43 \( 1 - 4392 T^{2} + 15655580 T^{4} - 32786290584 T^{6} + 70340912004102 T^{8} - 32786290584 p^{4} T^{10} + 15655580 p^{8} T^{12} - 4392 p^{12} T^{14} + p^{16} T^{16} \)
47 \( 1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 108976927256 p^{4} T^{10} + 34041564 p^{8} T^{12} - 6696 p^{12} T^{14} + p^{16} T^{16} \)
53 \( ( 1 - 24 T + 6108 T^{2} - 242920 T^{3} + 18930726 T^{4} - 242920 p^{2} T^{5} + 6108 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
59 \( 1 - 21392 T^{2} + 213726428 T^{4} - 1316320178544 T^{6} + 5495654048256518 T^{8} - 1316320178544 p^{4} T^{10} + 213726428 p^{8} T^{12} - 21392 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 96 T + 12728 T^{2} + 895872 T^{3} + 70065870 T^{4} + 895872 p^{2} T^{5} + 12728 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 - 16200 T^{2} + 120433884 T^{4} - 596093100152 T^{6} + 2634880977729030 T^{8} - 596093100152 p^{4} T^{10} + 120433884 p^{8} T^{12} - 16200 p^{12} T^{14} + p^{16} T^{16} \)
71 \( 1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 1991232124728 p^{4} T^{10} + 247457180 p^{8} T^{12} - 21000 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 136 T + 21660 T^{2} - 1811512 T^{3} + 170528390 T^{4} - 1811512 p^{2} T^{5} + 21660 p^{4} T^{6} - 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( 1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 1851543245752 p^{4} T^{10} + 289540636 p^{8} T^{12} - 26248 p^{12} T^{14} + p^{16} T^{16} \)
83 \( 1 - 19280 T^{2} + 168996572 T^{4} - 1078781742000 T^{6} + 6945567152070790 T^{8} - 1078781742000 p^{4} T^{10} + 168996572 p^{8} T^{12} - 19280 p^{12} T^{14} + p^{16} T^{16} \)
89 \( ( 1 + 40 T + 25916 T^{2} + 837912 T^{3} + 285914566 T^{4} + 837912 p^{2} T^{5} + 25916 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 7260024 p^{2} T^{5} + 52364 p^{4} T^{6} - 264 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−5.33306535825978581483637634575, −5.03800970056913530524778985981, −4.94385049692587675553021475238, −4.74812777397152279311185585249, −4.66008621705535079831659915351, −4.59489274929040809600377695679, −4.18440325178984026885294169897, −4.06514139371546035048262565926, −4.01417257710490480911557626573, −3.95141754454743559180683873301, −3.51682571060593413834509512078, −3.41450533786963782326992307115, −3.32536055898489011179326849833, −3.32376393142946992140980011745, −3.29861522692108960745638425725, −2.40456534071269030521606587998, −2.32006634762498395378840636344, −2.30104743066743134171992057560, −1.92057028481631718850800641211, −1.79837948263342960039456953692, −1.55485861189045533252588261315, −1.50098197212991132308958801179, −0.885762572433443073667195869933, −0.73647826206391503167501006961, −0.22975822131839937082282203411, 0.22975822131839937082282203411, 0.73647826206391503167501006961, 0.885762572433443073667195869933, 1.50098197212991132308958801179, 1.55485861189045533252588261315, 1.79837948263342960039456953692, 1.92057028481631718850800641211, 2.30104743066743134171992057560, 2.32006634762498395378840636344, 2.40456534071269030521606587998, 3.29861522692108960745638425725, 3.32376393142946992140980011745, 3.32536055898489011179326849833, 3.41450533786963782326992307115, 3.51682571060593413834509512078, 3.95141754454743559180683873301, 4.01417257710490480911557626573, 4.06514139371546035048262565926, 4.18440325178984026885294169897, 4.59489274929040809600377695679, 4.66008621705535079831659915351, 4.74812777397152279311185585249, 4.94385049692587675553021475238, 5.03800970056913530524778985981, 5.33306535825978581483637634575

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

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