Properties

Label 16-260e8-1.1-c1e8-0-2
Degree $16$
Conductor $2.088\times 10^{19}$
Sign $1$
Analytic cond. $345.146$
Root an. cond. $1.44087$
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  + 2·3-s − 2·5-s − 8·7-s + 2·9-s + 14·11-s + 6·13-s − 4·15-s − 2·19-s − 16·21-s + 22·23-s − 4·25-s − 2·27-s − 6·31-s + 28·33-s + 16·35-s − 20·37-s + 12·39-s − 8·41-s + 14·43-s − 4·45-s + 16·47-s + 16·49-s − 8·53-s − 28·55-s − 4·57-s − 2·59-s − 12·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 3.02·7-s + 2/3·9-s + 4.22·11-s + 1.66·13-s − 1.03·15-s − 0.458·19-s − 3.49·21-s + 4.58·23-s − 4/5·25-s − 0.384·27-s − 1.07·31-s + 4.87·33-s + 2.70·35-s − 3.28·37-s + 1.92·39-s − 1.24·41-s + 2.13·43-s − 0.596·45-s + 2.33·47-s + 16/7·49-s − 1.09·53-s − 3.77·55-s − 0.529·57-s − 0.260·59-s − 1.53·61-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(345.146\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.348483124\)
\(L(\frac12)\) \(\approx\) \(2.348483124\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 2 T + 8 T^{2} + 22 T^{3} + 38 T^{4} + 22 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13 \( 1 - 6 T + 4 T^{2} - 34 T^{3} + 366 T^{4} - 34 p T^{5} + 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
good3 \( 1 - 2 T + 2 T^{2} + 2 T^{3} - 4 T^{4} - 2 T^{5} + 14 T^{6} + 26 T^{7} - 86 T^{8} + 26 p T^{9} + 14 p^{2} T^{10} - 2 p^{3} T^{11} - 4 p^{4} T^{12} + 2 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \)
7 \( ( 1 + 4 T + 16 T^{2} + 8 p T^{3} + 170 T^{4} + 8 p^{2} T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 - 14 T + 98 T^{2} - 486 T^{3} + 2120 T^{4} - 9034 T^{5} + 36814 T^{6} - 138306 T^{7} + 477778 T^{8} - 138306 p T^{9} + 36814 p^{2} T^{10} - 9034 p^{3} T^{11} + 2120 p^{4} T^{12} - 486 p^{5} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 8 T^{3} - 116 T^{4} + 216 T^{5} + 32 T^{6} + 2224 T^{7} + 100710 T^{8} + 2224 p T^{9} + 32 p^{2} T^{10} + 216 p^{3} T^{11} - 116 p^{4} T^{12} - 8 p^{5} T^{13} + p^{8} T^{16} \)
19 \( 1 + 2 T + 2 T^{2} - 10 T^{3} - 584 T^{4} - 294 T^{5} + 630 T^{6} + 14718 T^{7} + 297906 T^{8} + 14718 p T^{9} + 630 p^{2} T^{10} - 294 p^{3} T^{11} - 584 p^{4} T^{12} - 10 p^{5} T^{13} + 2 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 22 T + 242 T^{2} - 2022 T^{3} + 15212 T^{4} - 101978 T^{5} + 606454 T^{6} - 3329826 T^{7} + 16823818 T^{8} - 3329826 p T^{9} + 606454 p^{2} T^{10} - 101978 p^{3} T^{11} + 15212 p^{4} T^{12} - 2022 p^{5} T^{13} + 242 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 76 T^{2} + 4540 T^{4} - 184020 T^{6} + 6139446 T^{8} - 184020 p^{2} T^{10} + 4540 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 + 6 T + 18 T^{2} + 142 T^{3} + 1144 T^{4} + 5034 T^{5} + 19694 T^{6} + 210370 T^{7} + 2234226 T^{8} + 210370 p T^{9} + 19694 p^{2} T^{10} + 5034 p^{3} T^{11} + 1144 p^{4} T^{12} + 142 p^{5} T^{13} + 18 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 + 10 T + 160 T^{2} + 1006 T^{3} + 8914 T^{4} + 1006 p T^{5} + 160 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 2172 T^{4} + 16928 T^{5} + 81408 T^{6} + 405608 T^{7} + 1063750 T^{8} + 405608 p T^{9} + 81408 p^{2} T^{10} + 16928 p^{3} T^{11} + 2172 p^{4} T^{12} + 176 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 14 T + 98 T^{2} - 10 p T^{3} - 2020 T^{4} + 30806 T^{5} - 140874 T^{6} + 92190 T^{7} + 3625258 T^{8} + 92190 p T^{9} - 140874 p^{2} T^{10} + 30806 p^{3} T^{11} - 2020 p^{4} T^{12} - 10 p^{6} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 20 T^{2} - 244 T^{3} + 2906 T^{4} - 244 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 + 8 T + 32 T^{2} - 504 T^{3} + 476 T^{4} + 50968 T^{5} + 519520 T^{6} + 102360 T^{7} - 12998234 T^{8} + 102360 p T^{9} + 519520 p^{2} T^{10} + 50968 p^{3} T^{11} + 476 p^{4} T^{12} - 504 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 2 T + 2 T^{2} + 70 T^{3} + 104 p T^{4} + 12666 T^{5} + 15510 T^{6} + 711918 T^{7} + 32447346 T^{8} + 711918 p T^{9} + 15510 p^{2} T^{10} + 12666 p^{3} T^{11} + 104 p^{5} T^{12} + 70 p^{5} T^{13} + 2 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
61 \( ( 1 + 6 T + 76 T^{2} - 278 T^{3} + 378 T^{4} - 278 p T^{5} + 76 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 436 T^{2} + 85948 T^{4} - 10230860 T^{6} + 821257334 T^{8} - 10230860 p^{2} T^{10} + 85948 p^{4} T^{12} - 436 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 + 22 T + 242 T^{2} + 2058 T^{3} + 14096 T^{4} + 3350 T^{5} - 1219850 T^{6} - 18442086 T^{7} - 179382110 T^{8} - 18442086 p T^{9} - 1219850 p^{2} T^{10} + 3350 p^{3} T^{11} + 14096 p^{4} T^{12} + 2058 p^{5} T^{13} + 242 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 52 T^{2} + 12748 T^{4} - 440492 T^{6} + 86661878 T^{8} - 440492 p^{2} T^{10} + 12748 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 292 T^{2} + 39172 T^{4} - 3359068 T^{6} + 252263350 T^{8} - 3359068 p^{2} T^{10} + 39172 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 4 T + 260 T^{2} + 1104 T^{3} + 29490 T^{4} + 1104 p T^{5} + 260 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 4 T + 8 T^{2} + 204 T^{3} - 9364 T^{4} - 35164 T^{5} - 44936 T^{6} - 55860 T^{7} + 65266150 T^{8} - 55860 p T^{9} - 44936 p^{2} T^{10} - 35164 p^{3} T^{11} - 9364 p^{4} T^{12} + 204 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 - 308 T^{2} + 55252 T^{4} - 7051276 T^{6} + 732653078 T^{8} - 7051276 p^{2} T^{10} + 55252 p^{4} T^{12} - 308 p^{6} T^{14} + p^{8} T^{16} \)
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

−5.58321566290035513017820445532, −5.37808340228172125384630060373, −4.98446737279916076492806754911, −4.77698559149156221559018094474, −4.75274026925413542664891637643, −4.56393421131696774890927956546, −4.39658377525730543503274699743, −4.30366321704302333469496237144, −3.95320427450859188881803574119, −3.80390578277389051854511736659, −3.74310438887295275044806330205, −3.52081220619482615988185832059, −3.51788027276266402832845621621, −3.31799565748255747947840172617, −3.31359197570091348417989344262, −3.16024367078306557408821261049, −2.78657306168973595053760686670, −2.77545441417593371549840029210, −2.40320754048800707495135634217, −1.94073800252152979210726248221, −1.83495712836157118606621709261, −1.40878721037335173620005217430, −1.28684528469693205093053778135, −1.17147077433232441609616278211, −0.43402105123047897583324596768, 0.43402105123047897583324596768, 1.17147077433232441609616278211, 1.28684528469693205093053778135, 1.40878721037335173620005217430, 1.83495712836157118606621709261, 1.94073800252152979210726248221, 2.40320754048800707495135634217, 2.77545441417593371549840029210, 2.78657306168973595053760686670, 3.16024367078306557408821261049, 3.31359197570091348417989344262, 3.31799565748255747947840172617, 3.51788027276266402832845621621, 3.52081220619482615988185832059, 3.74310438887295275044806330205, 3.80390578277389051854511736659, 3.95320427450859188881803574119, 4.30366321704302333469496237144, 4.39658377525730543503274699743, 4.56393421131696774890927956546, 4.75274026925413542664891637643, 4.77698559149156221559018094474, 4.98446737279916076492806754911, 5.37808340228172125384630060373, 5.58321566290035513017820445532

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

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