Properties

Label 16-5e32-1.1-c1e8-0-3
Degree $16$
Conductor $2.328\times 10^{22}$
Sign $1$
Analytic cond. $384819.$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 10·9-s − 4·11-s + 5·13-s − 16-s − 15·17-s + 10·19-s − 15·23-s + 15·27-s + 15·29-s + 31-s − 20·33-s + 5·37-s + 25·39-s − 9·41-s + 15·47-s − 5·48-s + 35·49-s − 75·51-s − 35·53-s + 50·57-s + 15·59-s + 6·61-s − 5·64-s − 75·69-s − 29·71-s − 10·73-s + ⋯
L(s)  = 1  + 2.88·3-s + 10/3·9-s − 1.20·11-s + 1.38·13-s − 1/4·16-s − 3.63·17-s + 2.29·19-s − 3.12·23-s + 2.88·27-s + 2.78·29-s + 0.179·31-s − 3.48·33-s + 0.821·37-s + 4.00·39-s − 1.40·41-s + 2.18·47-s − 0.721·48-s + 5·49-s − 10.5·51-s − 4.80·53-s + 6.62·57-s + 1.95·59-s + 0.768·61-s − 5/8·64-s − 9.02·69-s − 3.44·71-s − 1.17·73-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(5^{32}\)
Sign: $1$
Analytic conductor: \(384819.\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.455501360\)
\(L(\frac12)\) \(\approx\) \(2.455501360\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + T^{4} + 5 T^{6} - 5 T^{7} - 9 T^{8} - 5 p T^{9} + 5 p^{2} T^{10} + p^{4} T^{12} + p^{8} T^{16} \)
3 \( 1 - 5 T + 5 p T^{2} - 40 T^{3} + 32 p T^{4} - 205 T^{5} + 430 T^{6} - 850 T^{7} + 1531 T^{8} - 850 p T^{9} + 430 p^{2} T^{10} - 205 p^{3} T^{11} + 32 p^{5} T^{12} - 40 p^{5} T^{13} + 5 p^{7} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
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 \( 1 - 5 T + 40 T^{2} - 200 T^{3} + 956 T^{4} - 4155 T^{5} + 15835 T^{6} - 61940 T^{7} + 217296 T^{8} - 61940 p T^{9} + 15835 p^{2} T^{10} - 4155 p^{3} T^{11} + 956 p^{4} T^{12} - 200 p^{5} T^{13} + 40 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 15 T + 120 T^{2} + 570 T^{3} + 1786 T^{4} + 4065 T^{5} + 17855 T^{6} + 123410 T^{7} + 652356 T^{8} + 123410 p T^{9} + 17855 p^{2} T^{10} + 4065 p^{3} T^{11} + 1786 p^{4} T^{12} + 570 p^{5} T^{13} + 120 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 10 T + 12 T^{2} + 150 T^{3} - 197 T^{4} - 1210 T^{5} - 5166 T^{6} + 23900 T^{7} + 31275 T^{8} + 23900 p T^{9} - 5166 p^{2} T^{10} - 1210 p^{3} T^{11} - 197 p^{4} T^{12} + 150 p^{5} T^{13} + 12 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 + 15 T + 155 T^{2} + 1270 T^{3} + 8906 T^{4} + 55565 T^{5} + 314590 T^{6} + 1661560 T^{7} + 8221031 T^{8} + 1661560 p T^{9} + 314590 p^{2} T^{10} + 55565 p^{3} T^{11} + 8906 p^{4} T^{12} + 1270 p^{5} T^{13} + 155 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 15 T + 92 T^{2} - 380 T^{3} + 1868 T^{4} - 5065 T^{5} - 16861 T^{6} + 71400 T^{7} + 212600 T^{8} + 71400 p T^{9} - 16861 p^{2} T^{10} - 5065 p^{3} T^{11} + 1868 p^{4} T^{12} - 380 p^{5} T^{13} + 92 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - T - 65 T^{2} - 150 T^{3} + 860 T^{4} + 15887 T^{5} + 66068 T^{6} - 319870 T^{7} - 3557825 T^{8} - 319870 p T^{9} + 66068 p^{2} T^{10} + 15887 p^{3} T^{11} + 860 p^{4} T^{12} - 150 p^{5} T^{13} - 65 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 - 5 T + 75 T^{2} + 360 T^{3} - 994 T^{4} + 42345 T^{5} + 70860 T^{6} + 186740 T^{7} + 11831221 T^{8} + 186740 p T^{9} + 70860 p^{2} T^{10} + 42345 p^{3} T^{11} - 994 p^{4} T^{12} + 360 p^{5} T^{13} + 75 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 9 T - 20 T^{2} - 520 T^{3} - 1320 T^{4} + 22927 T^{5} + 153103 T^{6} - 256460 T^{7} - 5290960 T^{8} - 256460 p T^{9} + 153103 p^{2} T^{10} + 22927 p^{3} T^{11} - 1320 p^{4} T^{12} - 520 p^{5} T^{13} - 20 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
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 \( 1 - 15 T + 120 T^{2} - 960 T^{3} + 9636 T^{4} - 80865 T^{5} + 573845 T^{6} - 3818880 T^{7} + 26386476 T^{8} - 3818880 p T^{9} + 573845 p^{2} T^{10} - 80865 p^{3} T^{11} + 9636 p^{4} T^{12} - 960 p^{5} T^{13} + 120 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 35 T + 655 T^{2} + 8020 T^{3} + 67266 T^{4} + 345785 T^{5} + 173700 T^{6} - 16131620 T^{7} - 171061659 T^{8} - 16131620 p T^{9} + 173700 p^{2} T^{10} + 345785 p^{3} T^{11} + 67266 p^{4} T^{12} + 8020 p^{5} T^{13} + 655 p^{6} T^{14} + 35 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 15 T + 102 T^{2} - 840 T^{3} + 9468 T^{4} - 65115 T^{5} + 401879 T^{6} - 4632450 T^{7} + 44690100 T^{8} - 4632450 p T^{9} + 401879 p^{2} T^{10} - 65115 p^{3} T^{11} + 9468 p^{4} T^{12} - 840 p^{5} T^{13} + 102 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 6 T + 130 T^{2} - 135 T^{3} + 11205 T^{4} + 15672 T^{5} + 635993 T^{6} + 3243465 T^{7} + 39830270 T^{8} + 3243465 p T^{9} + 635993 p^{2} T^{10} + 15672 p^{3} T^{11} + 11205 p^{4} T^{12} - 135 p^{5} T^{13} + 130 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 130 T^{2} + 460 T^{3} + 6651 T^{4} + 59800 T^{5} + 247100 T^{6} + 6229960 T^{7} + 6681701 T^{8} + 6229960 p T^{9} + 247100 p^{2} T^{10} + 59800 p^{3} T^{11} + 6651 p^{4} T^{12} + 460 p^{5} T^{13} + 130 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 + 29 T + 320 T^{2} + 1480 T^{3} - 1630 T^{4} - 56363 T^{5} - 3597 T^{6} + 7019210 T^{7} + 88563540 T^{8} + 7019210 p T^{9} - 3597 p^{2} T^{10} - 56363 p^{3} T^{11} - 1630 p^{4} T^{12} + 1480 p^{5} T^{13} + 320 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 10 T + 220 T^{2} + 2525 T^{3} + 28471 T^{4} + 303710 T^{5} + 2569015 T^{6} + 24707575 T^{7} + 202533026 T^{8} + 24707575 p T^{9} + 2569015 p^{2} T^{10} + 303710 p^{3} T^{11} + 28471 p^{4} T^{12} + 2525 p^{5} T^{13} + 220 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 10 T + 112 T^{2} - 20 T^{3} - 2177 T^{4} - 102490 T^{5} - 216986 T^{6} - 1018300 T^{7} + 47464475 T^{8} - 1018300 p T^{9} - 216986 p^{2} T^{10} - 102490 p^{3} T^{11} - 2177 p^{4} T^{12} - 20 p^{5} T^{13} + 112 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 15 T + 155 T^{2} - 60 T^{3} - 11364 T^{4} - 155385 T^{5} - 467570 T^{6} + 4330590 T^{7} + 97045931 T^{8} + 4330590 p T^{9} - 467570 p^{2} T^{10} - 155385 p^{3} T^{11} - 11364 p^{4} T^{12} - 60 p^{5} T^{13} + 155 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 40 T + 562 T^{2} - 2590 T^{3} - 7557 T^{4} + 41760 T^{5} + 1856844 T^{6} - 44856200 T^{7} + 566687025 T^{8} - 44856200 p T^{9} + 1856844 p^{2} T^{10} + 41760 p^{3} T^{11} - 7557 p^{4} T^{12} - 2590 p^{5} T^{13} + 562 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 10 T + 75 T^{2} - 1560 T^{3} + 3371 T^{4} + 6540 T^{5} + 92385 T^{6} + 7411990 T^{7} - 78500144 T^{8} + 7411990 p T^{9} + 92385 p^{2} T^{10} + 6540 p^{3} T^{11} + 3371 p^{4} T^{12} - 1560 p^{5} T^{13} + 75 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.62465838329321984475709419802, −4.36137820216969175601908239965, −4.32307877921599315094205450459, −4.28459882328894573276294730170, −4.14201888295524261085303884440, −3.81590858635048340934948757747, −3.57368688620196898718683324028, −3.54989422416281736010848682107, −3.46593408972737583667434810120, −3.39911681572691594578327376107, −3.31134679401585691267017034318, −2.86196695945995320566698408567, −2.74302123884510714470608240525, −2.61023497879931567621194619115, −2.55938411614520548804773964398, −2.45577024393013796194616213774, −2.38151009724420803912576958671, −2.33292268133123587224604046697, −1.94481090798091498943819559218, −1.71873469978833849989652064776, −1.51693799396447120878339073007, −1.28086116453640222572151095141, −1.02431063576327737339719259524, −0.76196441026879066274146465946, −0.15459992459829442774747433099, 0.15459992459829442774747433099, 0.76196441026879066274146465946, 1.02431063576327737339719259524, 1.28086116453640222572151095141, 1.51693799396447120878339073007, 1.71873469978833849989652064776, 1.94481090798091498943819559218, 2.33292268133123587224604046697, 2.38151009724420803912576958671, 2.45577024393013796194616213774, 2.55938411614520548804773964398, 2.61023497879931567621194619115, 2.74302123884510714470608240525, 2.86196695945995320566698408567, 3.31134679401585691267017034318, 3.39911681572691594578327376107, 3.46593408972737583667434810120, 3.54989422416281736010848682107, 3.57368688620196898718683324028, 3.81590858635048340934948757747, 4.14201888295524261085303884440, 4.28459882328894573276294730170, 4.32307877921599315094205450459, 4.36137820216969175601908239965, 4.62465838329321984475709419802

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

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