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Properties

Label 16-2100e8-1.1-c0e8-0-5
Degree $16$
Conductor $3.782\times 10^{26}$
Sign $1$
Analytic cond. $1.45549$
Root an. cond. $1.02373$
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  − 12·31-s + 12·61-s + 81-s − 4·121-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 + 239-s + 241-s + ⋯
L(s)  = 1  − 12·31-s + 12·61-s + 81-s − 4·121-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 + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.45549\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074648846\)
\(L(\frac12)\) \(\approx\) \(1.074648846\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 + T^{2} )^{8} \)
31 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 - T^{4} + T^{8} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \)
67 \( ( 1 - T^{4} + T^{8} )^{2} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
79 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
97 \( ( 1 - T^{4} + T^{8} )^{2} \)
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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

−3.97480310746387872597649415265, −3.97426534813684829724170638642, −3.80151191636721812563174948801, −3.76329922708029081988276305432, −3.56485381761721820845231299097, −3.56154508626485508948789706783, −3.45327380785973644971106557594, −3.21008688704602477792723705882, −3.20010691258553470009044565321, −2.96156606003781770572298266331, −2.71434838345466428875317360185, −2.65531428191338931458688699375, −2.47788830730569797093290159977, −2.39614075720857589713401689299, −2.07361228374567090277296386066, −2.03757011849520878481465676673, −2.01497922518014752785080926093, −1.84530573513515084467263473055, −1.78317008364065344304475895942, −1.59210099542109706442567642959, −1.56723806494348369525921877677, −1.20973692382680506961616251657, −0.77019140704183156745807313154, −0.58972089032508073720877745366, −0.55337398647452010738487467046, 0.55337398647452010738487467046, 0.58972089032508073720877745366, 0.77019140704183156745807313154, 1.20973692382680506961616251657, 1.56723806494348369525921877677, 1.59210099542109706442567642959, 1.78317008364065344304475895942, 1.84530573513515084467263473055, 2.01497922518014752785080926093, 2.03757011849520878481465676673, 2.07361228374567090277296386066, 2.39614075720857589713401689299, 2.47788830730569797093290159977, 2.65531428191338931458688699375, 2.71434838345466428875317360185, 2.96156606003781770572298266331, 3.20010691258553470009044565321, 3.21008688704602477792723705882, 3.45327380785973644971106557594, 3.56154508626485508948789706783, 3.56485381761721820845231299097, 3.76329922708029081988276305432, 3.80151191636721812563174948801, 3.97426534813684829724170638642, 3.97480310746387872597649415265

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

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