Properties

Label 16-300e8-1.1-c1e8-0-9
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1084.39$
Root an. cond. $1.54774$
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  − 16-s − 16·41-s + 48·61-s − 2·81-s − 32·101-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  − 1/4·16-s − 2.49·41-s + 6.14·61-s − 2/9·81-s − 3.18·101-s − 2.18·121-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 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1084.39\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.487240385\)
\(L(\frac12)\) \(\approx\) \(2.487240385\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{4} + p^{4} T^{8} \)
3 \( ( 1 + T^{4} )^{2} \)
5 \( 1 \)
good7 \( ( 1 + 2 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \)
13 \( ( 1 - 334 T^{4} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + p^{2} T^{4} )^{4} \)
19 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 158 T^{4} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 622 T^{4} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 2 T + p T^{2} )^{8} \)
43 \( ( 1 - 3214 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 6 T + p T^{2} )^{8} \)
67 \( ( 1 - 8878 T^{4} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - p T^{2} )^{8} \)
73 \( ( 1 - 9502 T^{4} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 13294 T^{4} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 12094 T^{4} + p^{4} T^{8} )^{2} \)
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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

−5.24033285927500502702530747982, −5.19113952404350885894387436936, −5.16944596390723789737226246894, −5.04005629367846542259151802536, −4.62188967479394308618022846899, −4.51072559065223491151326086084, −4.29864171522189442654944847211, −4.20997994512210031092797214015, −3.89748684666363939275992693397, −3.81327161103452755568936856090, −3.78882890641502550285714123177, −3.74548443100461685988092369663, −3.41890987961110218561738125590, −3.10535575240065756426573400207, −2.96486803733508353378127532663, −2.87460730411740271355271987780, −2.49344010133412599717401419337, −2.46974784227322696232601674249, −2.38318431600931970056410301824, −1.86993810611148617767534413032, −1.76988193963928689605102938317, −1.51836927040862045277958455948, −1.36346394600829757414452701235, −0.63511398975853039823259255291, −0.57158506166021333835700289376, 0.57158506166021333835700289376, 0.63511398975853039823259255291, 1.36346394600829757414452701235, 1.51836927040862045277958455948, 1.76988193963928689605102938317, 1.86993810611148617767534413032, 2.38318431600931970056410301824, 2.46974784227322696232601674249, 2.49344010133412599717401419337, 2.87460730411740271355271987780, 2.96486803733508353378127532663, 3.10535575240065756426573400207, 3.41890987961110218561738125590, 3.74548443100461685988092369663, 3.78882890641502550285714123177, 3.81327161103452755568936856090, 3.89748684666363939275992693397, 4.20997994512210031092797214015, 4.29864171522189442654944847211, 4.51072559065223491151326086084, 4.62188967479394308618022846899, 5.04005629367846542259151802536, 5.16944596390723789737226246894, 5.19113952404350885894387436936, 5.24033285927500502702530747982

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

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