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Properties

Label 16-3871e8-1.1-c0e8-0-0
Degree $16$
Conductor $5.042\times 10^{28}$
Sign $1$
Analytic cond. $194.018$
Root an. cond. $1.38992$
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  + 2·2-s + 3·4-s + 2·8-s − 4·9-s − 2·11-s + 16-s − 8·18-s − 4·22-s − 2·23-s − 25-s − 2·32-s − 12·36-s − 6·44-s − 4·46-s − 2·50-s − 4·64-s − 2·67-s − 8·72-s − 4·79-s + 6·81-s − 4·88-s − 6·92-s + 8·99-s − 3·100-s + 3·121-s + 127-s − 6·128-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 2·8-s − 4·9-s − 2·11-s + 16-s − 8·18-s − 4·22-s − 2·23-s − 25-s − 2·32-s − 12·36-s − 6·44-s − 4·46-s − 2·50-s − 4·64-s − 2·67-s − 8·72-s − 4·79-s + 6·81-s − 4·88-s − 6·92-s + 8·99-s − 3·100-s + 3·121-s + 127-s − 6·128-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 79^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 79^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16} \cdot 79^{8}\)
Sign: $1$
Analytic conductor: \(194.018\)
Root analytic conductor: \(1.38992\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{16} \cdot 79^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043284098\)
\(L(\frac12)\) \(\approx\) \(1.043284098\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( ( 1 + T + T^{2} )^{4} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
3 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
5 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
11 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
19 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
23 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T )^{8}( 1 + T )^{8} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
47 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
53 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
83 \( ( 1 + T^{2} )^{8} \)
89 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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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

−3.70180779874214079730385429738, −3.58763230335673824686041887096, −3.32557449006868940272880548115, −3.26018600923007544471120074205, −3.24622909123191452871798952236, −3.15188964138775866271236113610, −3.03788841619853340569810308471, −2.94482706713681432402261453056, −2.76465293565438876849385135586, −2.69710203150533570807574634396, −2.69255934713490485608361994091, −2.60763642691824637302905423416, −2.40620082046900922729110816750, −2.38355283344323639850196272887, −2.08618437011516377030660491232, −1.91412241554188185707797258094, −1.88508075035991915843235837557, −1.76763429142546766449174187679, −1.70700681439116891386812644720, −1.53224890649912415382425593397, −1.34122615845573105812783629936, −0.955241862206047644328935932438, −0.69555244571411610459861753690, −0.33290528062476913509838912157, −0.31705660321478611486902155074, 0.31705660321478611486902155074, 0.33290528062476913509838912157, 0.69555244571411610459861753690, 0.955241862206047644328935932438, 1.34122615845573105812783629936, 1.53224890649912415382425593397, 1.70700681439116891386812644720, 1.76763429142546766449174187679, 1.88508075035991915843235837557, 1.91412241554188185707797258094, 2.08618437011516377030660491232, 2.38355283344323639850196272887, 2.40620082046900922729110816750, 2.60763642691824637302905423416, 2.69255934713490485608361994091, 2.69710203150533570807574634396, 2.76465293565438876849385135586, 2.94482706713681432402261453056, 3.03788841619853340569810308471, 3.15188964138775866271236113610, 3.24622909123191452871798952236, 3.26018600923007544471120074205, 3.32557449006868940272880548115, 3.58763230335673824686041887096, 3.70180779874214079730385429738

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

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