Properties

Label 16-570e8-1.1-c1e8-0-5
Degree $16$
Conductor $1.114\times 10^{22}$
Sign $1$
Analytic cond. $184168.$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 4·9-s + 10·16-s + 32·19-s + 4·25-s − 16·36-s + 32·49-s − 64·61-s − 20·64-s − 128·76-s − 6·81-s − 16·100-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 40·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 128·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 2·4-s + 4/3·9-s + 5/2·16-s + 7.34·19-s + 4/5·25-s − 8/3·36-s + 32/7·49-s − 8.19·61-s − 5/2·64-s − 14.6·76-s − 2/3·81-s − 8/5·100-s + 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.46·169-s + 9.78·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(184168.\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.836663331\)
\(L(\frac12)\) \(\approx\) \(4.836663331\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
5 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
19 \( ( 1 - 8 T + p T^{2} )^{4} \)
good7 \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + 8 T + p T^{2} )^{8} \)
67 \( ( 1 + p T^{2} )^{8} \)
71 \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{4} \)
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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

−4.75020795196865043886193082713, −4.58317267789085601599603807991, −4.57442960229952385474681392151, −4.24772602455232353809365655064, −4.21140715351182110625618874436, −3.95535799108867139064277337333, −3.87033497310690517431515260208, −3.75859410216273521225286326916, −3.64302162365838835193787385006, −3.29926079496753318872317968923, −3.15253493923282993385964218687, −3.04860567638519943019597844695, −3.03858777808618800227591470711, −2.97401590229859143539686600756, −2.90165344209995032257333272853, −2.62233891280913395759714076567, −2.04522298927654473501922214227, −2.00209811176355755722668978789, −1.82701818561583760460637274292, −1.43698726985165335190892075035, −1.28120733755846271928916825873, −1.15227538880699956886332753492, −0.966176085487300710750832794509, −0.789438381233177665965792283585, −0.42999619596698241405942853346, 0.42999619596698241405942853346, 0.789438381233177665965792283585, 0.966176085487300710750832794509, 1.15227538880699956886332753492, 1.28120733755846271928916825873, 1.43698726985165335190892075035, 1.82701818561583760460637274292, 2.00209811176355755722668978789, 2.04522298927654473501922214227, 2.62233891280913395759714076567, 2.90165344209995032257333272853, 2.97401590229859143539686600756, 3.03858777808618800227591470711, 3.04860567638519943019597844695, 3.15253493923282993385964218687, 3.29926079496753318872317968923, 3.64302162365838835193787385006, 3.75859410216273521225286326916, 3.87033497310690517431515260208, 3.95535799108867139064277337333, 4.21140715351182110625618874436, 4.24772602455232353809365655064, 4.57442960229952385474681392151, 4.58317267789085601599603807991, 4.75020795196865043886193082713

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

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