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Properties

Label 16-50e16-1.1-c0e8-0-1
Degree $16$
Conductor $1.526\times 10^{27}$
Sign $1$
Analytic cond. $5.87187$
Root an. cond. $1.11698$
Motivic weight $0$
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  + 4-s − 3·9-s − 6·29-s − 3·36-s − 4·41-s + 2·49-s + 6·61-s + 6·81-s − 6·89-s − 4·101-s + 4·109-s − 6·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4-s − 3·9-s − 6·29-s − 3·36-s − 4·41-s + 2·49-s + 6·61-s + 6·81-s − 6·89-s − 4·101-s + 4·109-s − 6·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(5.87187\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{32} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6376584810\)
\(L(\frac12)\) \(\approx\) \(0.6376584810\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 \)
good3 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
29 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
41 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
47 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
89 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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

−3.84054596958444014590883947637, −3.79816719399189894255585496450, −3.65284084308167011222067309429, −3.65121571868478375625952231858, −3.64672147612338442872376753403, −3.20257953901204852852976298966, −3.08708582382879308065376025631, −3.03359652452544854717038050729, −2.99770992761247854593624583089, −2.86916960390537221918179305969, −2.78129991147965456166618915443, −2.76788593266052022347166623852, −2.40053026532362934309339420900, −2.18350695079351917998037033690, −2.17229308819596054034662148630, −2.10422275818471476070073347601, −1.96785565724303448412911428536, −1.75891071309678734283969457490, −1.74926181396743575189722097491, −1.68780907719614261726436948462, −1.41813181501329240009208764361, −1.07601507551550642549937349332, −0.844240041775444926857334628517, −0.55465601012455008581468849267, −0.32500793612336343149295070688, 0.32500793612336343149295070688, 0.55465601012455008581468849267, 0.844240041775444926857334628517, 1.07601507551550642549937349332, 1.41813181501329240009208764361, 1.68780907719614261726436948462, 1.74926181396743575189722097491, 1.75891071309678734283969457490, 1.96785565724303448412911428536, 2.10422275818471476070073347601, 2.17229308819596054034662148630, 2.18350695079351917998037033690, 2.40053026532362934309339420900, 2.76788593266052022347166623852, 2.78129991147965456166618915443, 2.86916960390537221918179305969, 2.99770992761247854593624583089, 3.03359652452544854717038050729, 3.08708582382879308065376025631, 3.20257953901204852852976298966, 3.64672147612338442872376753403, 3.65121571868478375625952231858, 3.65284084308167011222067309429, 3.79816719399189894255585496450, 3.84054596958444014590883947637

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

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