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Properties

Label 16-475e8-1.1-c1e8-0-1
Degree $16$
Conductor $2.591\times 10^{21}$
Sign $1$
Analytic cond. $42831.7$
Root an. cond. $1.94753$
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  + 4·9-s + 8·11-s − 6·16-s − 8·19-s − 8·29-s + 8·31-s + 32·41-s + 8·49-s + 40·61-s + 4·64-s − 40·71-s + 32·79-s + 4·81-s − 8·89-s + 32·99-s + 24·101-s − 8·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s − 24·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 4/3·9-s + 2.41·11-s − 3/2·16-s − 1.83·19-s − 1.48·29-s + 1.43·31-s + 4.99·41-s + 8/7·49-s + 5.12·61-s + 1/2·64-s − 4.74·71-s + 3.60·79-s + 4/9·81-s − 0.847·89-s + 3.21·99-s + 2.38·101-s − 0.766·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(42831.7\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.520897735\)
\(L(\frac12)\) \(\approx\) \(6.520897735\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( ( 1 + T )^{8} \)
good2 \( 1 + 3 p T^{4} - p^{2} T^{6} + 17 T^{8} - p^{4} T^{10} + 3 p^{5} T^{12} + p^{8} T^{16} \)
3 \( 1 - 4 T^{2} + 4 p T^{4} - 20 p T^{6} + 214 T^{8} - 20 p^{3} T^{10} + 4 p^{5} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 - 4 T + 4 T^{2} - 20 T^{3} + 102 T^{4} - 20 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )( 1 + 4 T + 4 T^{2} + 20 T^{3} + 102 T^{4} + 20 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} ) \)
11 \( ( 1 - 4 T + 28 T^{2} - 100 T^{3} + 422 T^{4} - 100 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 4 p T^{2} + 1420 T^{4} - 27916 T^{6} + 417622 T^{8} - 27916 p^{2} T^{10} + 1420 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
17 \( 1 - 56 T^{2} + 1180 T^{4} - 9864 T^{6} + 38854 T^{8} - 9864 p^{2} T^{10} + 1180 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 72 T^{2} + 3324 T^{4} - 112888 T^{6} + 2951366 T^{8} - 112888 p^{2} T^{10} + 3324 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 4 T + 84 T^{2} + 332 T^{3} + 3238 T^{4} + 332 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 4 T + 44 T^{2} + 140 T^{3} + 166 T^{4} + 140 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 212 T^{2} + 20748 T^{4} - 1267308 T^{6} + 54701334 T^{8} - 1267308 p^{2} T^{10} + 20748 p^{4} T^{12} - 212 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 16 T + 220 T^{2} - 1936 T^{3} + 14438 T^{4} - 1936 p T^{5} + 220 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 296 T^{2} + 40092 T^{4} - 3238872 T^{6} + 170514342 T^{8} - 3238872 p^{2} T^{10} + 40092 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 104 T^{2} + 7100 T^{4} - 7720 p T^{6} + 15690118 T^{8} - 7720 p^{3} T^{10} + 7100 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 324 T^{2} + 49836 T^{4} - 4722364 T^{6} + 301404374 T^{8} - 4722364 p^{2} T^{10} + 49836 p^{4} T^{12} - 324 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 172 T^{2} + 224 T^{3} + 13142 T^{4} + 224 p T^{5} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 20 T + 300 T^{2} - 2972 T^{3} + 26502 T^{4} - 2972 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 228 T^{2} + 17356 T^{4} + 3172 T^{6} - 58243754 T^{8} + 3172 p^{2} T^{10} + 17356 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 20 T + 316 T^{2} + 3236 T^{3} + 30566 T^{4} + 3236 p T^{5} + 316 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 312 T^{2} + 52060 T^{4} - 5937416 T^{6} + 497988358 T^{8} - 5937416 p^{2} T^{10} + 52060 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 16 T + 348 T^{2} - 3312 T^{3} + 40646 T^{4} - 3312 p T^{5} + 348 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 520 T^{2} + 127324 T^{4} - 19097976 T^{6} + 1915966182 T^{8} - 19097976 p^{2} T^{10} + 127324 p^{4} T^{12} - 520 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 4 T + 212 T^{2} + 1244 T^{3} + 22134 T^{4} + 1244 p T^{5} + 212 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 324 T^{2} + 47308 T^{4} - 4899740 T^{6} + 476587702 T^{8} - 4899740 p^{2} T^{10} + 47308 p^{4} T^{12} - 324 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.77859203708049122191010818547, −4.56952923420103925989509375143, −4.55348236871424453517401474087, −4.44431220327236172372108792776, −4.32172223239521193372700528828, −3.98843348495488712191951193556, −3.92900924194938324985376089738, −3.92336696996106343873576540469, −3.86138305339665772217166338092, −3.65925379067751753885779145820, −3.60271660837948958690660395966, −3.30186181523681964635845890534, −2.96093173373823314272777587970, −2.68593218287033978848223639891, −2.60797115770489139217578981990, −2.47655391463815517244080790620, −2.47182541616849707509172488546, −2.10346460820102597972094642394, −2.03021432719410596123109253626, −1.74274653898676888450612137925, −1.51109000446287994664237236763, −1.27161693456061227521917898448, −1.06652075441511720032289296724, −0.61140150303042469998615000199, −0.60555899744086569538930769682, 0.60555899744086569538930769682, 0.61140150303042469998615000199, 1.06652075441511720032289296724, 1.27161693456061227521917898448, 1.51109000446287994664237236763, 1.74274653898676888450612137925, 2.03021432719410596123109253626, 2.10346460820102597972094642394, 2.47182541616849707509172488546, 2.47655391463815517244080790620, 2.60797115770489139217578981990, 2.68593218287033978848223639891, 2.96093173373823314272777587970, 3.30186181523681964635845890534, 3.60271660837948958690660395966, 3.65925379067751753885779145820, 3.86138305339665772217166338092, 3.92336696996106343873576540469, 3.92900924194938324985376089738, 3.98843348495488712191951193556, 4.32172223239521193372700528828, 4.44431220327236172372108792776, 4.55348236871424453517401474087, 4.56952923420103925989509375143, 4.77859203708049122191010818547

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

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