Properties

Label 16-384e8-1.1-c3e8-0-2
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $6.94349\times 10^{10}$
Root an. cond. $4.75990$
Motivic weight $3$
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  − 60·9-s − 640·23-s − 392·25-s + 984·49-s − 640·71-s − 1.23e3·73-s + 1.24e3·81-s + 5.71e3·97-s + 6.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.06e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 3.84e4·207-s + ⋯
L(s)  = 1  − 2.22·9-s − 5.80·23-s − 3.13·25-s + 2.86·49-s − 1.06·71-s − 1.97·73-s + 1.70·81-s + 5.97·97-s + 4.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.12·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 12.8·207-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.94349\times 10^{10}\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.339300794\)
\(L(\frac12)\) \(\approx\) \(2.339300794\)
\(L(\frac{5}{2})\) 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 \( ( 1 + 10 p T^{2} + p^{6} T^{4} )^{2} \)
good5 \( ( 1 + 196 T^{2} + 774 p^{2} T^{4} + 196 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
7 \( ( 1 - 492 T^{2} + 102278 T^{4} - 492 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
11 \( ( 1 - 3108 T^{2} + 5613974 T^{4} - 3108 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
13 \( ( 1 - 4532 T^{2} + 10573590 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
17 \( ( 1 - 6916 T^{2} + 54727878 T^{4} - 6916 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
19 \( ( 1 - 3588 T^{2} + 29873654 T^{4} - 3588 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
23 \( ( 1 + 160 T + 29390 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
29 \( ( 1 + 28900 T^{2} + 841043958 T^{4} + 28900 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
31 \( ( 1 - 112780 T^{2} + 4945356198 T^{4} - 112780 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( ( 1 - 40148 T^{2} + 1484010870 T^{4} - 40148 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
41 \( ( 1 - 198884 T^{2} + 18596196390 T^{4} - 198884 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
43 \( ( 1 + 174108 T^{2} + 16916734358 T^{4} + 174108 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
47 \( ( 1 + 121630 T^{2} + p^{6} T^{4} )^{4} \)
53 \( ( 1 + 133508 T^{2} - 4558541226 T^{4} + 133508 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 - 408658 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - 129524 T^{2} + 106016792790 T^{4} - 129524 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
67 \( ( 1 + 690876 T^{2} + 254657072438 T^{4} + 690876 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
71 \( ( 1 + 160 T - 117778 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
73 \( ( 1 + 308 T + 608214 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
79 \( ( 1 - 257100 T^{2} + 501931869158 T^{4} - 257100 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
83 \( ( 1 - 1153988 T^{2} + 779081408118 T^{4} - 1153988 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
89 \( ( 1 - 1957732 T^{2} + 1892598888294 T^{4} - 1957732 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
97 \( ( 1 - 1428 T + 2249126 T^{2} - 1428 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
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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

−4.51085962044791509918175844042, −4.33602897597073461059352141375, −4.26751901753234692874637441823, −4.02488526074733755609183165296, −3.89308984050703027655006004632, −3.84408653621172773166330098432, −3.61796578053825311822453518478, −3.55989685177603178916149582489, −3.37516432675778798063781439050, −3.13130538175012750713476397562, −3.09047918561115116620530451857, −2.69859829696648727389772793429, −2.62441086887628283933217215185, −2.35230738961978979277341729680, −2.28706216497373339454775517235, −2.05047045044042402139240085462, −1.98384941106184680361136379435, −1.81493841636855457344196936608, −1.72631015693474768333407764614, −1.40947890281369833194396881850, −1.03882729035575698699368683502, −0.57520644305495793112511196890, −0.46346802451615248199005524492, −0.32578687765351167420783356335, −0.26453521424781961298245148793, 0.26453521424781961298245148793, 0.32578687765351167420783356335, 0.46346802451615248199005524492, 0.57520644305495793112511196890, 1.03882729035575698699368683502, 1.40947890281369833194396881850, 1.72631015693474768333407764614, 1.81493841636855457344196936608, 1.98384941106184680361136379435, 2.05047045044042402139240085462, 2.28706216497373339454775517235, 2.35230738961978979277341729680, 2.62441086887628283933217215185, 2.69859829696648727389772793429, 3.09047918561115116620530451857, 3.13130538175012750713476397562, 3.37516432675778798063781439050, 3.55989685177603178916149582489, 3.61796578053825311822453518478, 3.84408653621172773166330098432, 3.89308984050703027655006004632, 4.02488526074733755609183165296, 4.26751901753234692874637441823, 4.33602897597073461059352141375, 4.51085962044791509918175844042

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

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