Properties

Label 16-48e16-1.1-c1e8-0-13
Degree $16$
Conductor $7.941\times 10^{26}$
Sign $1$
Analytic cond. $1.31243\times 10^{10}$
Root an. cond. $4.28923$
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  − 8·13-s + 56·37-s + 48·49-s − 40·61-s + 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 2.21·13-s + 9.20·37-s + 48/7·49-s − 5.12·61-s + 0.766·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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 + 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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.31243\times 10^{10}\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.772203947\)
\(L(\frac12)\) \(\approx\) \(5.772203947\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
7 \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 238 T^{4} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \)
17 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + 434 T^{4} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 12 T + p T^{2} )^{4}( 1 - 2 T + p T^{2} )^{4} \)
41 \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 1198 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
59 \( ( 1 - 6478 T^{4} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 4078 T^{4} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 11278 T^{4} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + p T^{2} )^{8} \)
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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

−3.84330315839403386843539097244, −3.80449850334678184012602791907, −3.45088615407426926406090551141, −3.16635070409653421668687653711, −3.16344413941354898621403275097, −3.13916853697289742398110358409, −3.09613472086421004419943726993, −3.00376670248954057352045618489, −2.59800999935659401004634874796, −2.54586004195960141404283907529, −2.46460684292076548120397777403, −2.46157972166981223988065037473, −2.21616301020542578668076544150, −2.20939552478319932793719297743, −2.16315755230333119864710357551, −2.14646042968956249478068080592, −1.55016276352500378858196602241, −1.41265193476939621273408601493, −1.27417705220042437672595364847, −1.05723979810389542253047699728, −1.01330456036734425147560323510, −0.922822481754712890938729462596, −0.69231730655818857243605206551, −0.31790769441746445381781981961, −0.26474507335898672956310492280, 0.26474507335898672956310492280, 0.31790769441746445381781981961, 0.69231730655818857243605206551, 0.922822481754712890938729462596, 1.01330456036734425147560323510, 1.05723979810389542253047699728, 1.27417705220042437672595364847, 1.41265193476939621273408601493, 1.55016276352500378858196602241, 2.14646042968956249478068080592, 2.16315755230333119864710357551, 2.20939552478319932793719297743, 2.21616301020542578668076544150, 2.46157972166981223988065037473, 2.46460684292076548120397777403, 2.54586004195960141404283907529, 2.59800999935659401004634874796, 3.00376670248954057352045618489, 3.09613472086421004419943726993, 3.13916853697289742398110358409, 3.16344413941354898621403275097, 3.16635070409653421668687653711, 3.45088615407426926406090551141, 3.80449850334678184012602791907, 3.84330315839403386843539097244

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

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