Properties

Label 16-8470e8-1.1-c1e8-0-0
Degree $16$
Conductor $2.649\times 10^{31}$
Sign $1$
Analytic cond. $4.37808\times 10^{14}$
Root an. cond. $8.22394$
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  + 8·2-s + 36·4-s − 8·5-s − 8·7-s + 120·8-s − 8·9-s − 64·10-s − 13-s − 64·14-s + 330·16-s + 6·17-s − 64·18-s + 5·19-s − 288·20-s + 10·23-s + 36·25-s − 8·26-s − 288·28-s + 3·29-s − 8·31-s + 792·32-s + 48·34-s + 64·35-s − 288·36-s − 6·37-s + 40·38-s − 960·40-s + ⋯
L(s)  = 1  + 5.65·2-s + 18·4-s − 3.57·5-s − 3.02·7-s + 42.4·8-s − 8/3·9-s − 20.2·10-s − 0.277·13-s − 17.1·14-s + 82.5·16-s + 1.45·17-s − 15.0·18-s + 1.14·19-s − 64.3·20-s + 2.08·23-s + 36/5·25-s − 1.56·26-s − 54.4·28-s + 0.557·29-s − 1.43·31-s + 140.·32-s + 8.23·34-s + 10.8·35-s − 48·36-s − 0.986·37-s + 6.48·38-s − 151.·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(4.37808\times 10^{14}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 5^{8} \cdot 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(572.0603522\)
\(L(\frac12)\) \(\approx\) \(572.0603522\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{8} \)
5 \( ( 1 + T )^{8} \)
7 \( ( 1 + T )^{8} \)
11 \( 1 \)
good3 \( 1 + 8 T^{2} + 11 p T^{4} - 10 T^{5} + 110 T^{6} - 80 T^{7} + 341 T^{8} - 80 p T^{9} + 110 p^{2} T^{10} - 10 p^{3} T^{11} + 11 p^{5} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 + T + 17 T^{2} - 49 T^{3} + 162 T^{4} - 701 T^{5} + 2811 T^{6} - 10103 T^{7} + 23142 T^{8} - 10103 p T^{9} + 2811 p^{2} T^{10} - 701 p^{3} T^{11} + 162 p^{4} T^{12} - 49 p^{5} T^{13} + 17 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 6 T + 74 T^{2} - 364 T^{3} + 2517 T^{4} - 584 p T^{5} + 55138 T^{6} - 186138 T^{7} + 980057 T^{8} - 186138 p T^{9} + 55138 p^{2} T^{10} - 584 p^{4} T^{11} + 2517 p^{4} T^{12} - 364 p^{5} T^{13} + 74 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
19 \( 1 - 5 T + 46 T^{2} - 180 T^{3} + 82 p T^{4} - 7065 T^{5} + 45063 T^{6} - 159420 T^{7} + 887004 T^{8} - 159420 p T^{9} + 45063 p^{2} T^{10} - 7065 p^{3} T^{11} + 82 p^{5} T^{12} - 180 p^{5} T^{13} + 46 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 10 T + 148 T^{2} - 1130 T^{3} + 9268 T^{4} - 59090 T^{5} + 350700 T^{6} - 1945810 T^{7} + 9369686 T^{8} - 1945810 p T^{9} + 350700 p^{2} T^{10} - 59090 p^{3} T^{11} + 9268 p^{4} T^{12} - 1130 p^{5} T^{13} + 148 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 3 T + 119 T^{2} - 299 T^{3} + 6620 T^{4} - 15955 T^{5} + 253525 T^{6} - 618047 T^{7} + 7942994 T^{8} - 618047 p T^{9} + 253525 p^{2} T^{10} - 15955 p^{3} T^{11} + 6620 p^{4} T^{12} - 299 p^{5} T^{13} + 119 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 8 T + 128 T^{2} + 386 T^{3} + 4792 T^{4} - 1878 T^{5} + 153964 T^{6} - 83560 T^{7} + 6250342 T^{8} - 83560 p T^{9} + 153964 p^{2} T^{10} - 1878 p^{3} T^{11} + 4792 p^{4} T^{12} + 386 p^{5} T^{13} + 128 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 6 T + 166 T^{2} + 1112 T^{3} + 13056 T^{4} + 89704 T^{5} + 690310 T^{6} + 4474010 T^{7} + 28407366 T^{8} + 4474010 p T^{9} + 690310 p^{2} T^{10} + 89704 p^{3} T^{11} + 13056 p^{4} T^{12} + 1112 p^{5} T^{13} + 166 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 11 T + 124 T^{2} - 878 T^{3} + 6428 T^{4} - 22765 T^{5} + 100545 T^{6} + 298878 T^{7} - 1904064 T^{8} + 298878 p T^{9} + 100545 p^{2} T^{10} - 22765 p^{3} T^{11} + 6428 p^{4} T^{12} - 878 p^{5} T^{13} + 124 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 + 5 T + 156 T^{2} + 1112 T^{3} + 14252 T^{4} + 2209 p T^{5} + 958199 T^{6} + 5367826 T^{7} + 47078316 T^{8} + 5367826 p T^{9} + 958199 p^{2} T^{10} + 2209 p^{4} T^{11} + 14252 p^{4} T^{12} + 1112 p^{5} T^{13} + 156 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 15 T + 275 T^{2} + 2525 T^{3} + 28426 T^{4} + 213045 T^{5} + 2042525 T^{6} + 14025095 T^{7} + 114368906 T^{8} + 14025095 p T^{9} + 2042525 p^{2} T^{10} + 213045 p^{3} T^{11} + 28426 p^{4} T^{12} + 2525 p^{5} T^{13} + 275 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 16 T + 192 T^{2} + 1966 T^{3} + 21900 T^{4} + 193178 T^{5} + 1606988 T^{6} + 12594320 T^{7} + 98924158 T^{8} + 12594320 p T^{9} + 1606988 p^{2} T^{10} + 193178 p^{3} T^{11} + 21900 p^{4} T^{12} + 1966 p^{5} T^{13} + 192 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 9 T + 234 T^{2} + 2104 T^{3} + 34306 T^{4} + 269797 T^{5} + 3244791 T^{6} + 22625212 T^{7} + 226891572 T^{8} + 22625212 p T^{9} + 3244791 p^{2} T^{10} + 269797 p^{3} T^{11} + 34306 p^{4} T^{12} + 2104 p^{5} T^{13} + 234 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 32 T + 824 T^{2} - 14626 T^{3} + 222408 T^{4} - 2765794 T^{5} + 496748 p T^{6} - 284999148 T^{7} + 2388069542 T^{8} - 284999148 p T^{9} + 496748 p^{3} T^{10} - 2765794 p^{3} T^{11} + 222408 p^{4} T^{12} - 14626 p^{5} T^{13} + 824 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 - 33 T + 842 T^{2} - 15042 T^{3} + 229138 T^{4} - 2884273 T^{5} + 32190375 T^{6} - 311931366 T^{7} + 2722654900 T^{8} - 311931366 p T^{9} + 32190375 p^{2} T^{10} - 2884273 p^{3} T^{11} + 229138 p^{4} T^{12} - 15042 p^{5} T^{13} + 842 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 11 T + 425 T^{2} - 4759 T^{3} + 86016 T^{4} - 913881 T^{5} + 10870055 T^{6} - 102265729 T^{7} + 929735530 T^{8} - 102265729 p T^{9} + 10870055 p^{2} T^{10} - 913881 p^{3} T^{11} + 86016 p^{4} T^{12} - 4759 p^{5} T^{13} + 425 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 34 T + 752 T^{2} + 11904 T^{3} + 158025 T^{4} + 1775712 T^{5} + 18078418 T^{6} + 167305460 T^{7} + 1476137933 T^{8} + 167305460 p T^{9} + 18078418 p^{2} T^{10} + 1775712 p^{3} T^{11} + 158025 p^{4} T^{12} + 11904 p^{5} T^{13} + 752 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 31 T + 713 T^{2} - 10199 T^{3} + 119396 T^{4} - 957285 T^{5} + 6373271 T^{6} - 24224753 T^{7} + 156969274 T^{8} - 24224753 p T^{9} + 6373271 p^{2} T^{10} - 957285 p^{3} T^{11} + 119396 p^{4} T^{12} - 10199 p^{5} T^{13} + 713 p^{6} T^{14} - 31 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 - 50 T + 1544 T^{2} - 33386 T^{3} + 572973 T^{4} - 8071520 T^{5} + 98073518 T^{6} - 1045485168 T^{7} + 10042941577 T^{8} - 1045485168 p T^{9} + 98073518 p^{2} T^{10} - 8071520 p^{3} T^{11} + 572973 p^{4} T^{12} - 33386 p^{5} T^{13} + 1544 p^{6} T^{14} - 50 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - T + 504 T^{2} - 1162 T^{3} + 117472 T^{4} - 392061 T^{5} + 17106935 T^{6} - 63616750 T^{7} + 1769275296 T^{8} - 63616750 p T^{9} + 17106935 p^{2} T^{10} - 392061 p^{3} T^{11} + 117472 p^{4} T^{12} - 1162 p^{5} T^{13} + 504 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 4 T + 392 T^{2} + 1000 T^{3} + 71413 T^{4} + 145422 T^{5} + 9028218 T^{6} + 19286276 T^{7} + 945132769 T^{8} + 19286276 p T^{9} + 9028218 p^{2} T^{10} + 145422 p^{3} T^{11} + 71413 p^{4} T^{12} + 1000 p^{5} T^{13} + 392 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
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

−3.23646563288345857764497732961, −3.10083032428550460206026454233, −3.08316058595281667051443601452, −3.01394940324880800209783699245, −2.99062239401367938632965236246, −2.76219715856958530344917479469, −2.75112978452970531929764305336, −2.73095402420099297124503418686, −2.64921991915006968329801242433, −2.26639910097728168729042201693, −2.10418885573954133631208542428, −2.05984319739156743736786688927, −2.01951979470211914309428645931, −1.98494038726488819603554362848, −1.84342289878337857735896068073, −1.68116901702903111545204043288, −1.54301787602735412524392174222, −1.17615636695872939995265826848, −0.847047782546539297161638653282, −0.823221426125558794979795274433, −0.71248825117380874315896609022, −0.64115784489123138212963027147, −0.56133309910317678777462080526, −0.55923809348416851371711986409, −0.28039423432335704463269953369, 0.28039423432335704463269953369, 0.55923809348416851371711986409, 0.56133309910317678777462080526, 0.64115784489123138212963027147, 0.71248825117380874315896609022, 0.823221426125558794979795274433, 0.847047782546539297161638653282, 1.17615636695872939995265826848, 1.54301787602735412524392174222, 1.68116901702903111545204043288, 1.84342289878337857735896068073, 1.98494038726488819603554362848, 2.01951979470211914309428645931, 2.05984319739156743736786688927, 2.10418885573954133631208542428, 2.26639910097728168729042201693, 2.64921991915006968329801242433, 2.73095402420099297124503418686, 2.75112978452970531929764305336, 2.76219715856958530344917479469, 2.99062239401367938632965236246, 3.01394940324880800209783699245, 3.08316058595281667051443601452, 3.10083032428550460206026454233, 3.23646563288345857764497732961

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

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