Properties

Label 16-5e16-1.1-c1e8-0-0
Degree $16$
Conductor $152587890625$
Sign $1$
Analytic cond. $2.52195\times 10^{-6}$
Root an. cond. $0.446795$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 5·3-s + 10·4-s + 25·6-s − 5·8-s + 10·9-s − 4·11-s − 50·12-s − 5·13-s − 16·16-s − 10·17-s − 50·18-s − 5·19-s + 20·22-s + 5·23-s + 25·24-s − 5·25-s + 25·26-s − 5·27-s − 5·29-s − 9·31-s + 30·32-s + 20·33-s + 50·34-s + 100·36-s + 30·37-s + 25·38-s + ⋯
L(s)  = 1  − 3.53·2-s − 2.88·3-s + 5·4-s + 10.2·6-s − 1.76·8-s + 10/3·9-s − 1.20·11-s − 14.4·12-s − 1.38·13-s − 4·16-s − 2.42·17-s − 11.7·18-s − 1.14·19-s + 4.26·22-s + 1.04·23-s + 5.10·24-s − 25-s + 4.90·26-s − 0.962·27-s − 0.928·29-s − 1.61·31-s + 5.30·32-s + 3.48·33-s + 8.57·34-s + 50/3·36-s + 4.93·37-s + 4.05·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16}\)
Sign: $1$
Analytic conductor: \(2.52195\times 10^{-6}\)
Root analytic conductor: \(0.446795\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{25} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.003735116372\)
\(L(\frac12)\) \(\approx\) \(0.003735116372\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T^{2} - 4 p T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{8} \)
good2 \( 1 + 5 T + 15 T^{2} + 15 p T^{3} + 41 T^{4} + 15 p T^{5} - 5 p^{2} T^{6} - 55 p T^{7} - 199 T^{8} - 55 p^{2} T^{9} - 5 p^{4} T^{10} + 15 p^{4} T^{11} + 41 p^{4} T^{12} + 15 p^{6} T^{13} + 15 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
3 \( 1 + 5 T + 5 p T^{2} + 10 p T^{3} + 4 p^{2} T^{4} + 5 T^{5} - 40 p T^{6} - 400 T^{7} - 809 T^{8} - 400 p T^{9} - 40 p^{3} T^{10} + 5 p^{3} T^{11} + 4 p^{6} T^{12} + 10 p^{6} T^{13} + 5 p^{7} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - 5 p T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 7045 p^{2} T^{10} + 611 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T - 7 T^{2} - 36 T^{3} + 5 T^{4} - 36 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 5 T + 30 T^{2} + 60 T^{3} + 346 T^{4} + 655 T^{5} + 6335 T^{6} + 13320 T^{7} + 88856 T^{8} + 13320 p T^{9} + 6335 p^{2} T^{10} + 655 p^{3} T^{11} + 346 p^{4} T^{12} + 60 p^{5} T^{13} + 30 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 10 T + 90 T^{2} + 720 T^{3} + 4451 T^{4} + 26310 T^{5} + 136780 T^{6} + 638720 T^{7} + 2802941 T^{8} + 638720 p T^{9} + 136780 p^{2} T^{10} + 26310 p^{3} T^{11} + 4451 p^{4} T^{12} + 720 p^{5} T^{13} + 90 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 + 5 T - 8 T^{2} + 40 T^{3} + 878 T^{4} + 1705 T^{5} - 1861 T^{6} + 31550 T^{7} + 293380 T^{8} + 31550 p T^{9} - 1861 p^{2} T^{10} + 1705 p^{3} T^{11} + 878 p^{4} T^{12} + 40 p^{5} T^{13} - 8 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 5 T + 45 T^{2} + 100 T^{3} - 104 T^{4} + 7645 T^{5} + 18890 T^{6} + 8420 T^{7} + 1238691 T^{8} + 8420 p T^{9} + 18890 p^{2} T^{10} + 7645 p^{3} T^{11} - 104 p^{4} T^{12} + 100 p^{5} T^{13} + 45 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 5 T - 28 T^{2} + 140 T^{3} + 1268 T^{4} - 5045 T^{5} + 48219 T^{6} + 207000 T^{7} - 1738520 T^{8} + 207000 p T^{9} + 48219 p^{2} T^{10} - 5045 p^{3} T^{11} + 1268 p^{4} T^{12} + 140 p^{5} T^{13} - 28 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 9 T + 55 T^{2} + 390 T^{3} + 2980 T^{4} + 20297 T^{5} + 114748 T^{6} + 589990 T^{7} + 3582095 T^{8} + 589990 p T^{9} + 114748 p^{2} T^{10} + 20297 p^{3} T^{11} + 2980 p^{4} T^{12} + 390 p^{5} T^{13} + 55 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 30 T + 480 T^{2} - 5675 T^{3} + 56171 T^{4} - 489630 T^{5} + 3826535 T^{6} - 26904445 T^{7} + 171416106 T^{8} - 26904445 p T^{9} + 3826535 p^{2} T^{10} - 489630 p^{3} T^{11} + 56171 p^{4} T^{12} - 5675 p^{5} T^{13} + 480 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 4 T - 30 T^{2} - 240 T^{3} + 195 T^{4} + 18372 T^{5} + 77388 T^{6} - 230240 T^{7} - 1438395 T^{8} - 230240 p T^{9} + 77388 p^{2} T^{10} + 18372 p^{3} T^{11} + 195 p^{4} T^{12} - 240 p^{5} T^{13} - 30 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 5 p T^{2} + 22911 T^{4} - 1578205 T^{6} + 78597176 T^{8} - 1578205 p^{2} T^{10} + 22911 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \)
47 \( 1 + 110 T^{2} - 90 T^{3} + 4101 T^{4} - 9900 T^{5} - 3830 T^{6} - 910260 T^{7} - 5610889 T^{8} - 910260 p T^{9} - 3830 p^{2} T^{10} - 9900 p^{3} T^{11} + 4101 p^{4} T^{12} - 90 p^{5} T^{13} + 110 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 10 T + 100 T^{2} + 1625 T^{3} + 11531 T^{4} + 95310 T^{5} + 857875 T^{6} + 5635035 T^{7} + 42472426 T^{8} + 5635035 p T^{9} + 857875 p^{2} T^{10} + 95310 p^{3} T^{11} + 11531 p^{4} T^{12} + 1625 p^{5} T^{13} + 100 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 2 p T^{2} + 900 T^{3} + 3393 T^{4} - 96300 T^{5} + 615034 T^{6} + 2943900 T^{7} - 65890945 T^{8} + 2943900 p T^{9} + 615034 p^{2} T^{10} - 96300 p^{3} T^{11} + 3393 p^{4} T^{12} + 900 p^{5} T^{13} - 2 p^{7} T^{14} + p^{8} T^{16} \)
61 \( 1 + 9 T - 165 T^{2} - 1800 T^{3} + 7560 T^{4} + 154437 T^{5} + 526138 T^{6} - 4559670 T^{7} - 69838275 T^{8} - 4559670 p T^{9} + 526138 p^{2} T^{10} + 154437 p^{3} T^{11} + 7560 p^{4} T^{12} - 1800 p^{5} T^{13} - 165 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 20 T + 250 T^{2} - 2600 T^{3} + 26091 T^{4} - 241820 T^{5} + 2034700 T^{6} - 15361680 T^{7} + 117317461 T^{8} - 15361680 p T^{9} + 2034700 p^{2} T^{10} - 241820 p^{3} T^{11} + 26091 p^{4} T^{12} - 2600 p^{5} T^{13} + 250 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 6 T - 910 T^{3} + 4875 T^{4} + 34402 T^{5} + 474398 T^{6} - 3835500 T^{7} - 25760305 T^{8} - 3835500 p T^{9} + 474398 p^{2} T^{10} + 34402 p^{3} T^{11} + 4875 p^{4} T^{12} - 910 p^{5} T^{13} - 6 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 15 T + 195 T^{2} - 1340 T^{3} + 15786 T^{4} - 132465 T^{5} + 1900340 T^{6} - 200800 p T^{7} + 154250461 T^{8} - 200800 p^{2} T^{9} + 1900340 p^{2} T^{10} - 132465 p^{3} T^{11} + 15786 p^{4} T^{12} - 1340 p^{5} T^{13} + 195 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 15 T - 58 T^{2} + 2180 T^{3} - 7802 T^{4} - 148865 T^{5} + 1559169 T^{6} + 5584500 T^{7} - 169067020 T^{8} + 5584500 p T^{9} + 1559169 p^{2} T^{10} - 148865 p^{3} T^{11} - 7802 p^{4} T^{12} + 2180 p^{5} T^{13} - 58 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 45 T + 1115 T^{2} + 19890 T^{3} + 284376 T^{4} + 3450645 T^{5} + 448160 p T^{6} + 367888080 T^{7} + 3431240591 T^{8} + 367888080 p T^{9} + 448160 p^{3} T^{10} + 3450645 p^{3} T^{11} + 284376 p^{4} T^{12} + 19890 p^{5} T^{13} + 1115 p^{6} T^{14} + 45 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 + 25 T + 342 T^{2} + 5000 T^{3} + 78868 T^{4} + 928525 T^{5} + 9098049 T^{6} + 101226750 T^{7} + 1076434080 T^{8} + 101226750 p T^{9} + 9098049 p^{2} T^{10} + 928525 p^{3} T^{11} + 78868 p^{4} T^{12} + 5000 p^{5} T^{13} + 342 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 60 T + 1830 T^{2} + 35660 T^{3} + 467711 T^{4} + 3770460 T^{5} + 6583460 T^{6} - 292271720 T^{7} - 4451764059 T^{8} - 292271720 p T^{9} + 6583460 p^{2} T^{10} + 3770460 p^{3} T^{11} + 467711 p^{4} T^{12} + 35660 p^{5} T^{13} + 1830 p^{6} T^{14} + 60 p^{7} T^{15} + p^{8} T^{16} \)
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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

−9.264606005725786684581250817764, −8.701519957482883251445840148556, −8.700756789123480086234663064578, −8.587410407659643003356611381504, −8.313327649596639150128601111549, −8.022729140610343966777265765300, −8.000658794287396241187871279826, −7.73124663888318285965963720408, −7.37214446289042285892258381094, −7.31387626590244159416471837223, −7.04203274766215539700169588614, −6.85744330449983430802479407420, −6.54132635344714138363853998282, −6.45852707771919679444432037317, −5.82385696664051564495606249014, −5.66095007107135267912137327348, −5.61736961642789035495983217468, −5.52198619498244300122709572032, −4.97705277057159681438592943029, −4.73810456658168035860164939240, −4.36048904079619467511194010037, −4.20351592571445873243353432084, −3.98048819631383695223672152000, −2.51905553356217036605454282958, −2.48425883173590507807753867406, 2.48425883173590507807753867406, 2.51905553356217036605454282958, 3.98048819631383695223672152000, 4.20351592571445873243353432084, 4.36048904079619467511194010037, 4.73810456658168035860164939240, 4.97705277057159681438592943029, 5.52198619498244300122709572032, 5.61736961642789035495983217468, 5.66095007107135267912137327348, 5.82385696664051564495606249014, 6.45852707771919679444432037317, 6.54132635344714138363853998282, 6.85744330449983430802479407420, 7.04203274766215539700169588614, 7.31387626590244159416471837223, 7.37214446289042285892258381094, 7.73124663888318285965963720408, 8.000658794287396241187871279826, 8.022729140610343966777265765300, 8.313327649596639150128601111549, 8.587410407659643003356611381504, 8.700756789123480086234663064578, 8.701519957482883251445840148556, 9.264606005725786684581250817764

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

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