↑

Properties

Label 16-1170e8-1.1-c1e8-0-6
Degree $16$
Conductor $3.511\times 10^{24}$
Sign $1$
Analytic cond. $5.80368\times 10^{7}$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·4-s − 3·5-s + 5·7-s − 12·10-s + 3·11-s + 4·13-s + 20·14-s − 15·16-s + 15·17-s + 9·19-s − 18·20-s + 12·22-s + 6·23-s + 7·25-s + 16·26-s + 30·28-s + 3·29-s − 24·32-s + 60·34-s − 15·35-s + 20·37-s + 36·38-s − 21·41-s + 18·43-s + 18·44-s + 24·46-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s − 1.34·5-s + 1.88·7-s − 3.79·10-s + 0.904·11-s + 1.10·13-s + 5.34·14-s − 3.75·16-s + 3.63·17-s + 2.06·19-s − 4.02·20-s + 2.55·22-s + 1.25·23-s + 7/5·25-s + 3.13·26-s + 5.66·28-s + 0.557·29-s − 4.24·32-s + 10.2·34-s − 2.53·35-s + 3.28·37-s + 5.83·38-s − 3.27·41-s + 2.74·43-s + 2.71·44-s + 3.53·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(5.80368\times 10^{7}\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(25.53190216\)
\(L(\frac12)\) \(\approx\) \(25.53190216\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 + 3 T + 2 T^{2} - 3 p T^{3} - 54 T^{4} - 3 p^{2} T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13 \( 1 - 4 T + 28 T^{2} - 118 T^{3} + 523 T^{4} - 118 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( 1 - 5 T + 6 T^{2} + 41 T^{3} - 199 T^{4} + 432 T^{5} - 74 T^{6} - 3242 T^{7} + 13092 T^{8} - 3242 p T^{9} - 74 p^{2} T^{10} + 432 p^{3} T^{11} - 199 p^{4} T^{12} + 41 p^{5} T^{13} + 6 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 - 3 T + 28 T^{2} - 75 T^{3} + 397 T^{4} - 1512 T^{5} + 5050 T^{6} - 23220 T^{7} + 57412 T^{8} - 23220 p T^{9} + 5050 p^{2} T^{10} - 1512 p^{3} T^{11} + 397 p^{4} T^{12} - 75 p^{5} T^{13} + 28 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 15 T + 151 T^{2} - 1140 T^{3} + 7387 T^{4} - 42657 T^{5} + 221164 T^{6} - 1045737 T^{7} + 4488412 T^{8} - 1045737 p T^{9} + 221164 p^{2} T^{10} - 42657 p^{3} T^{11} + 7387 p^{4} T^{12} - 1140 p^{5} T^{13} + 151 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 9 T + 94 T^{2} - 603 T^{3} + 205 p T^{4} - 18144 T^{5} + 93484 T^{6} - 373770 T^{7} + 1769680 T^{8} - 373770 p T^{9} + 93484 p^{2} T^{10} - 18144 p^{3} T^{11} + 205 p^{5} T^{12} - 603 p^{5} T^{13} + 94 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 6 T + 88 T^{2} - 456 T^{3} + 4126 T^{4} - 21306 T^{5} + 141472 T^{6} - 672282 T^{7} + 3633727 T^{8} - 672282 p T^{9} + 141472 p^{2} T^{10} - 21306 p^{3} T^{11} + 4126 p^{4} T^{12} - 456 p^{5} T^{13} + 88 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 3 T - 89 T^{2} + 108 T^{3} + 5023 T^{4} - 1659 T^{5} - 202094 T^{6} + 22485 T^{7} + 6307612 T^{8} + 22485 p T^{9} - 202094 p^{2} T^{10} - 1659 p^{3} T^{11} + 5023 p^{4} T^{12} + 108 p^{5} T^{13} - 89 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 188 T^{2} + 16924 T^{4} - 941624 T^{6} + 35207710 T^{8} - 941624 p^{2} T^{10} + 16924 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 20 T + 132 T^{2} - 484 T^{3} + 5861 T^{4} - 55470 T^{5} + 263104 T^{6} - 1941398 T^{7} + 16342164 T^{8} - 1941398 p T^{9} + 263104 p^{2} T^{10} - 55470 p^{3} T^{11} + 5861 p^{4} T^{12} - 484 p^{5} T^{13} + 132 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 21 T + 283 T^{2} + 2856 T^{3} + 23455 T^{4} + 158745 T^{5} + 910306 T^{6} + 5010921 T^{7} + 29380960 T^{8} + 5010921 p T^{9} + 910306 p^{2} T^{10} + 158745 p^{3} T^{11} + 23455 p^{4} T^{12} + 2856 p^{5} T^{13} + 283 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 18 T + 244 T^{2} - 2448 T^{3} + 19486 T^{4} - 115506 T^{5} + 569656 T^{6} - 2068326 T^{7} + 9076111 T^{8} - 2068326 p T^{9} + 569656 p^{2} T^{10} - 115506 p^{3} T^{11} + 19486 p^{4} T^{12} - 2448 p^{5} T^{13} + 244 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 + 3 T + 71 T^{2} + 156 T^{3} + 2160 T^{4} + 156 p T^{5} + 71 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 - 68 T^{2} + 7318 T^{4} - 456764 T^{6} + 29553859 T^{8} - 456764 p^{2} T^{10} + 7318 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 + 30 T + 580 T^{2} + 8400 T^{3} + 101470 T^{4} + 1072734 T^{5} + 10172152 T^{6} + 88229610 T^{7} + 703838287 T^{8} + 88229610 p T^{9} + 10172152 p^{2} T^{10} + 1072734 p^{3} T^{11} + 101470 p^{4} T^{12} + 8400 p^{5} T^{13} + 580 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 5 T + 27 T^{2} + 208 T^{3} + 2873 T^{4} + 16809 T^{5} + 25594 T^{6} + 1324415 T^{7} + 2164560 T^{8} + 1324415 p T^{9} + 25594 p^{2} T^{10} + 16809 p^{3} T^{11} + 2873 p^{4} T^{12} + 208 p^{5} T^{13} + 27 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
67 \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( 1 + 124 T^{2} + 3754 T^{4} + 20352 T^{5} + 240112 T^{6} + 4462848 T^{7} + 35638579 T^{8} + 4462848 p T^{9} + 240112 p^{2} T^{10} + 20352 p^{3} T^{11} + 3754 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 13 T + 178 T^{2} - 1549 T^{3} + 13924 T^{4} - 1549 p T^{5} + 178 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 2 T + 100 T^{2} + 28 T^{3} + 4702 T^{4} + 28 p T^{5} + 100 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 24 T + 410 T^{2} - 4374 T^{3} + 44538 T^{4} - 4374 p T^{5} + 410 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 39 T + 904 T^{2} - 15483 T^{3} + 210337 T^{4} - 2365440 T^{5} + 23189230 T^{6} - 211685526 T^{7} + 1943350048 T^{8} - 211685526 p T^{9} + 23189230 p^{2} T^{10} - 2365440 p^{3} T^{11} + 210337 p^{4} T^{12} - 15483 p^{5} T^{13} + 904 p^{6} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 4 T - 312 T^{2} - 244 T^{3} + 60230 T^{4} - 23286 T^{5} - 8035652 T^{6} + 909880 T^{7} + 827060415 T^{8} + 909880 p T^{9} - 8035652 p^{2} T^{10} - 23286 p^{3} T^{11} + 60230 p^{4} T^{12} - 244 p^{5} T^{13} - 312 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.12636455782982890808399401202, −4.04937254032562250258342765428, −3.86128305234849324814952467498, −3.72088722394256757474343120823, −3.64466619222434603006327369693, −3.61920740429055048376610833522, −3.51144811638149694300625871481, −3.25982906638136678332267315494, −3.22849685954295944790714592460, −3.18782027682405562041570404737, −3.08141445069800115294748202460, −3.04601669347377840768626578638, −2.58125808042253930416452679600, −2.45727548819684809481300814570, −2.35342864395694242614242475821, −2.23811581330248905797270036381, −2.05821163175796444210337462677, −1.86007336469338322999590052904, −1.31876187897860238378166933302, −1.23354752730937458170488031603, −1.09096764707361508429884282699, −1.08682725493193421819311801582, −1.08526716583600662850606112803, −0.841348094394824368203268719794, −0.22196383610002700850193461226, 0.22196383610002700850193461226, 0.841348094394824368203268719794, 1.08526716583600662850606112803, 1.08682725493193421819311801582, 1.09096764707361508429884282699, 1.23354752730937458170488031603, 1.31876187897860238378166933302, 1.86007336469338322999590052904, 2.05821163175796444210337462677, 2.23811581330248905797270036381, 2.35342864395694242614242475821, 2.45727548819684809481300814570, 2.58125808042253930416452679600, 3.04601669347377840768626578638, 3.08141445069800115294748202460, 3.18782027682405562041570404737, 3.22849685954295944790714592460, 3.25982906638136678332267315494, 3.51144811638149694300625871481, 3.61920740429055048376610833522, 3.64466619222434603006327369693, 3.72088722394256757474343120823, 3.86128305234849324814952467498, 4.04937254032562250258342765428, 4.12636455782982890808399401202

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

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