Properties

Label 16-1620e8-1.1-c2e8-0-5
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $1.44145\times 10^{13}$
Root an. cond. $6.64392$
Motivic weight $2$
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  + 96·19-s − 42·25-s + 152·31-s − 104·49-s + 280·61-s − 120·79-s + 592·109-s − 288·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 152·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 5.05·19-s − 1.67·25-s + 4.90·31-s − 2.12·49-s + 4.59·61-s − 1.51·79-s + 5.43·109-s − 2.38·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.899·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.44145\times 10^{13}\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(53.59272789\)
\(L(\frac12)\) \(\approx\) \(53.59272789\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + 42 T^{2} + 1139 T^{4} + 42 p^{4} T^{6} + p^{8} T^{8} \)
good7 \( ( 1 + 52 T^{2} + 303 T^{4} + 52 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 + 144 T^{2} + 6095 T^{4} + 144 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 76 T^{2} - 22785 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{4} \)
19 \( ( 1 - 12 T + p^{2} T^{2} )^{8} \)
23 \( ( 1 - 42 p T^{2} + 1235 p^{2} T^{4} - 42 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 1610 T^{2} + 1884819 T^{4} + 1610 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 38 T + 483 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 2692 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 1440 T^{2} - 752161 T^{4} - 1440 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 902 T^{2} - 2605197 T^{4} - 902 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 + 1470 T^{2} - 2718781 T^{4} + 1470 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 + p^{2} T^{2} )^{8} \)
59 \( ( 1 - p^{4} T^{4} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 70 T + 1179 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
67 \( ( 1 - 2798 T^{2} - 12322317 T^{4} - 2798 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 + 4030 T^{2} + p^{4} T^{4} )^{4} \)
73 \( ( 1 - 10474 T^{2} + p^{4} T^{4} )^{4} \)
79 \( ( 1 + 30 T - 5341 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 + 4254 T^{2} - 29361805 T^{4} + 4254 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 14784 T^{2} + p^{4} T^{4} )^{4} \)
97 \( ( 1 + 9802 T^{2} + 7549923 T^{4} + 9802 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−3.79499118972699040799104465979, −3.53812925462567466032525516457, −3.43584120695676983686940701872, −3.26431311959624513699171379486, −3.22434597582822635688556808585, −3.22419862735086994775839847240, −3.01307147942563627788342620934, −2.81065831424944422154419004166, −2.72796628090727211821543204044, −2.63854890995332164458987902074, −2.53619467986412609031087062997, −2.45062412640099605973355419350, −2.25083008138266189984366220974, −1.87519522960586846762291479927, −1.78573601149878640828767913137, −1.65905580590479644345216927533, −1.60989358505051742037958348957, −1.45901529037273705395838906854, −1.10685167613841384126529542385, −1.06044373382658988661719329003, −0.881142415728753404656334120808, −0.64196217447151302228840326111, −0.53563941042292312692510881476, −0.48683700732759647862798945681, −0.40145754742198908905967184349, 0.40145754742198908905967184349, 0.48683700732759647862798945681, 0.53563941042292312692510881476, 0.64196217447151302228840326111, 0.881142415728753404656334120808, 1.06044373382658988661719329003, 1.10685167613841384126529542385, 1.45901529037273705395838906854, 1.60989358505051742037958348957, 1.65905580590479644345216927533, 1.78573601149878640828767913137, 1.87519522960586846762291479927, 2.25083008138266189984366220974, 2.45062412640099605973355419350, 2.53619467986412609031087062997, 2.63854890995332164458987902074, 2.72796628090727211821543204044, 2.81065831424944422154419004166, 3.01307147942563627788342620934, 3.22419862735086994775839847240, 3.22434597582822635688556808585, 3.26431311959624513699171379486, 3.43584120695676983686940701872, 3.53812925462567466032525516457, 3.79499118972699040799104465979

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

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