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Properties

Label 16-549e8-1.1-c0e8-0-0
Degree $16$
Conductor $8.252\times 10^{21}$
Sign $1$
Analytic cond. $3.17567\times 10^{-5}$
Root an. cond. $0.523437$
Motivic weight $0$
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  − 2·7-s + 16-s − 2·25-s + 2·31-s + 2·37-s + 2·43-s − 3·49-s − 8·61-s + 2·67-s − 4·73-s + 2·79-s − 2·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·175-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·7-s + 16-s − 2·25-s + 2·31-s + 2·37-s + 2·43-s − 3·49-s − 8·61-s + 2·67-s − 4·73-s + 2·79-s − 2·112-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·175-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 61^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 61^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 61^{8}\)
Sign: $1$
Analytic conductor: \(3.17567\times 10^{-5}\)
Root analytic conductor: \(0.523437\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 61^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2505474565\)
\(L(\frac12)\) \(\approx\) \(0.2505474565\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( ( 1 + T )^{8} \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( ( 1 + T^{4} )^{4} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
29 \( ( 1 + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
47 \( ( 1 + T^{2} )^{8} \)
53 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
59 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
71 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
79 \( ( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
97 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−4.92516533156510579888051646450, −4.85699504619533326594364658631, −4.63841818050621946604294296639, −4.40534660223070033500020779481, −4.39304557212664067788213787014, −4.28887035213218109920875051655, −4.16974294208111651978328848980, −4.16529777043254907931158009597, −4.11213068037704310074193326724, −3.48543351108537923384160290058, −3.36846236977531076451229214751, −3.36595866685993249042469367590, −3.29195380152340305332433350727, −3.02620962040006135228647496596, −2.99511867751246029088592546939, −2.90578548678019531080031131571, −2.84138331558685522520427612352, −2.57027792774344133181378244258, −2.11437412218788355750976206406, −1.95698940469754546678342796155, −1.84511066277277509537899650493, −1.79769722080992501028462806948, −1.40629819124618092391968174503, −1.12466932511876079540437530956, −0.77698781937104220804901960370, 0.77698781937104220804901960370, 1.12466932511876079540437530956, 1.40629819124618092391968174503, 1.79769722080992501028462806948, 1.84511066277277509537899650493, 1.95698940469754546678342796155, 2.11437412218788355750976206406, 2.57027792774344133181378244258, 2.84138331558685522520427612352, 2.90578548678019531080031131571, 2.99511867751246029088592546939, 3.02620962040006135228647496596, 3.29195380152340305332433350727, 3.36595866685993249042469367590, 3.36846236977531076451229214751, 3.48543351108537923384160290058, 4.11213068037704310074193326724, 4.16529777043254907931158009597, 4.16974294208111651978328848980, 4.28887035213218109920875051655, 4.39304557212664067788213787014, 4.40534660223070033500020779481, 4.63841818050621946604294296639, 4.85699504619533326594364658631, 4.92516533156510579888051646450

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

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