Properties

Label 16-21e16-1.1-c1e8-0-0
Degree $16$
Conductor $1.431\times 10^{21}$
Sign $1$
Analytic cond. $23644.2$
Root an. cond. $1.87654$
Motivic weight $1$
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  + 4·4-s + 10·16-s − 32·37-s − 32·43-s + 36·64-s − 32·79-s − 32·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s − 128·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2·4-s + 5/2·16-s − 5.26·37-s − 4.87·43-s + 9/2·64-s − 3.60·79-s − 3.06·109-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s − 9.75·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(23644.2\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3300486965\)
\(L(\frac12)\) \(\approx\) \(0.3300486965\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
5 \( ( 1 + 48 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 16 T^{2} + 64 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 4 T^{2} + 694 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 44 T^{2} + 2374 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 108 T^{2} + 7302 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 156 T^{2} + 13014 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 176 T^{2} + 14128 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 260 T^{2} + 26854 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 240 T^{2} + 24480 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 192 T^{2} + 23136 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 336 T^{2} + 46464 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.87804183958878609864324921893, −4.78742408271253130936965406469, −4.66692034548636840498003141407, −4.56081943841721085116607636033, −4.43129745911099922223459928746, −4.07133810283170706612900237609, −3.96453924116737751355353249677, −3.94954358646271740796109979890, −3.50969728613556461161087510119, −3.40483890303602421570490644077, −3.33378652358355589080491716145, −3.29971236218686794797978573628, −3.21641956255137791994715397057, −3.06899434770365071138075860206, −2.67463167110657769592876196361, −2.59586479729940079319977276241, −2.40502404901435535113910720856, −1.96321025692432800132810724762, −1.90607441545801680421054824736, −1.82239806321550030655164306323, −1.71650265971139107293354797900, −1.61764564249852980721179987264, −1.10455842338892364057098247357, −0.895127659542059370612270288503, −0.085685749917432787358306129667, 0.085685749917432787358306129667, 0.895127659542059370612270288503, 1.10455842338892364057098247357, 1.61764564249852980721179987264, 1.71650265971139107293354797900, 1.82239806321550030655164306323, 1.90607441545801680421054824736, 1.96321025692432800132810724762, 2.40502404901435535113910720856, 2.59586479729940079319977276241, 2.67463167110657769592876196361, 3.06899434770365071138075860206, 3.21641956255137791994715397057, 3.29971236218686794797978573628, 3.33378652358355589080491716145, 3.40483890303602421570490644077, 3.50969728613556461161087510119, 3.94954358646271740796109979890, 3.96453924116737751355353249677, 4.07133810283170706612900237609, 4.43129745911099922223459928746, 4.56081943841721085116607636033, 4.66692034548636840498003141407, 4.78742408271253130936965406469, 4.87804183958878609864324921893

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

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