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Properties

Label 16-93e8-1.1-c1e8-0-0
Degree $16$
Conductor $5.596\times 10^{15}$
Sign $1$
Analytic cond. $0.0924869$
Root an. cond. $0.861747$
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  + 3·3-s − 4·4-s − 4·7-s + 3·9-s − 12·12-s + 9·13-s + 4·16-s + 7·19-s − 12·21-s − 20·25-s + 16·28-s − 4·31-s − 12·36-s + 9·37-s + 27·39-s + 44·43-s + 12·48-s + 9·49-s − 36·52-s + 21·57-s − 12·63-s + 16·67-s + 3·73-s − 60·75-s − 28·76-s − 35·79-s + 48·84-s + ⋯
L(s)  = 1  + 1.73·3-s − 2·4-s − 1.51·7-s + 9-s − 3.46·12-s + 2.49·13-s + 16-s + 1.60·19-s − 2.61·21-s − 4·25-s + 3.02·28-s − 0.718·31-s − 2·36-s + 1.47·37-s + 4.32·39-s + 6.70·43-s + 1.73·48-s + 9/7·49-s − 4.99·52-s + 2.78·57-s − 1.51·63-s + 1.95·67-s + 0.351·73-s − 6.92·75-s − 3.21·76-s − 3.93·79-s + 5.23·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(0.0924869\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7391011874\)
\(L(\frac12)\) \(\approx\) \(0.7391011874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
31 \( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} - 184 p T^{5} - 15 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
good2 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
5 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
7 \( ( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 55 p T^{5} + 18 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} ) \)
11 \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 7 T + 36 T^{2} - 161 T^{3} + 659 T^{4} - 161 p T^{5} + 36 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} + 44 p T^{5} - 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} ) \)
17 \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 208 p T^{5} + 45 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \)
23 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 10 T + 63 T^{2} - 260 T^{3} + 269 T^{4} - 260 p T^{5} + 63 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )( 1 + T - 36 T^{2} - 73 T^{3} + 1259 T^{4} - 73 p T^{5} - 36 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \)
41 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{5} T^{10} - p^{7} T^{14} + p^{8} T^{16} \)
43 \( ( 1 - 13 T + p T^{2} )^{4}( 1 + 8 T + 21 T^{2} - 176 T^{3} - 2311 T^{4} - 176 p T^{5} + 21 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} ) \)
47 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16} \)
59 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{5} T^{10} - p^{7} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 1036 p T^{5} + 135 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )( 1 + 14 T + 135 T^{2} + 1036 T^{3} + 6269 T^{4} + 1036 p T^{5} + 135 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} ) \)
67 \( ( 1 - 11 T + 54 T^{2} + 143 T^{3} - 5191 T^{4} + 143 p T^{5} + 54 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )( 1 - 5 T - 42 T^{2} + 545 T^{3} + 89 T^{4} + 545 p T^{5} - 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} ) \)
71 \( 1 - p T^{2} + p^{3} T^{6} - p^{4} T^{8} + p^{5} T^{10} - p^{7} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 10 T + 27 T^{2} + 460 T^{3} - 6571 T^{4} + 460 p T^{5} + 27 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )( 1 + 7 T - 24 T^{2} - 679 T^{3} - 3001 T^{4} - 679 p T^{5} - 24 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} ) \)
79 \( ( 1 + 13 T + p T^{2} )^{4}( 1 - 17 T + 210 T^{2} - 2227 T^{3} + 21269 T^{4} - 2227 p T^{5} + 210 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} ) \)
83 \( 1 + p T^{2} - p^{3} T^{6} - p^{4} T^{8} - p^{5} T^{10} + p^{7} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 5 T + p T^{2} )^{4}( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} - 845 p T^{5} - 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} ) \)
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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

−6.89258889516758012516794185462, −6.32936944027612822576010998756, −6.14923387345495093590831470243, −6.14404177466395507931258090812, −5.79424825936722170624041122562, −5.76574762956726733518220062836, −5.70979325045132626554403560203, −5.63839884395016422868469324249, −5.42634859090805473472869064111, −5.01823402559749398759468173287, −4.62180512863275309057174678269, −4.53150263395175964780550367247, −4.21899608228898718554282925570, −4.08730349124566849199642013513, −4.02817152785137470180630507046, −3.95735900805063109629136221088, −3.69064321658200426380421896390, −3.32923879114260490518111164281, −3.28446896734065664730281152784, −3.01025219821789969513365137599, −2.54668429802209106816778612373, −2.38344690633852038406427839166, −2.30832901970724047000910561847, −1.52666541951277348169336733726, −0.991100769029793808485667639067, 0.991100769029793808485667639067, 1.52666541951277348169336733726, 2.30832901970724047000910561847, 2.38344690633852038406427839166, 2.54668429802209106816778612373, 3.01025219821789969513365137599, 3.28446896734065664730281152784, 3.32923879114260490518111164281, 3.69064321658200426380421896390, 3.95735900805063109629136221088, 4.02817152785137470180630507046, 4.08730349124566849199642013513, 4.21899608228898718554282925570, 4.53150263395175964780550367247, 4.62180512863275309057174678269, 5.01823402559749398759468173287, 5.42634859090805473472869064111, 5.63839884395016422868469324249, 5.70979325045132626554403560203, 5.76574762956726733518220062836, 5.79424825936722170624041122562, 6.14404177466395507931258090812, 6.14923387345495093590831470243, 6.32936944027612822576010998756, 6.89258889516758012516794185462

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

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