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Properties

Label 16-341e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.828\times 10^{20}$
Sign $1$
Analytic cond. $7.03545\times 10^{-7}$
Root an. cond. $0.412530$
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  + 2·3-s − 2·4-s − 5-s + 3·9-s + 11-s − 4·12-s − 2·15-s + 16-s + 2·20-s − 3·23-s + 2·27-s − 4·31-s + 2·33-s − 6·36-s + 4·37-s − 2·44-s − 3·45-s − 3·47-s + 2·48-s + 49-s − 53-s − 55-s − 59-s + 4·60-s − 67-s − 6·69-s − 71-s + ⋯
L(s)  = 1  + 2·3-s − 2·4-s − 5-s + 3·9-s + 11-s − 4·12-s − 2·15-s + 16-s + 2·20-s − 3·23-s + 2·27-s − 4·31-s + 2·33-s − 6·36-s + 4·37-s − 2·44-s − 3·45-s − 3·47-s + 2·48-s + 49-s − 53-s − 55-s − 59-s + 4·60-s − 67-s − 6·69-s − 71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(11^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(7.03545\times 10^{-7}\)
Root analytic conductor: \(0.412530\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 11^{8} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1687066240\)
\(L(\frac12)\) \(\approx\) \(0.1687066240\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
31 \( ( 1 + T + T^{2} )^{4} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
47 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
97 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
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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

−5.42881904571923356394990334317, −5.22075209533499278869260106210, −4.91981947250966653096374003001, −4.72180629822626357837871862928, −4.71017628835552676561306985158, −4.68418667432380258451833795593, −4.39047882260428115452575234933, −4.35231551415535401904748027263, −4.25180349335114163320145476191, −4.18093819809721662668549555680, −3.81323234831385677883051680107, −3.74555542296501755636744039675, −3.57754543150164341534101601826, −3.46831563717061927600611900627, −3.40628111158166396043308822416, −3.40293645836101657418482748674, −3.03692593075440879267818183873, −2.63284368066158686864824576706, −2.54141004057693487306329685495, −2.29875502150060506174441279954, −1.94942748050430246236236596611, −1.92364984160243096044191606973, −1.84944040770812992943118842090, −1.44314604318679200764951611161, −1.14148092823919998699225731218, 1.14148092823919998699225731218, 1.44314604318679200764951611161, 1.84944040770812992943118842090, 1.92364984160243096044191606973, 1.94942748050430246236236596611, 2.29875502150060506174441279954, 2.54141004057693487306329685495, 2.63284368066158686864824576706, 3.03692593075440879267818183873, 3.40293645836101657418482748674, 3.40628111158166396043308822416, 3.46831563717061927600611900627, 3.57754543150164341534101601826, 3.74555542296501755636744039675, 3.81323234831385677883051680107, 4.18093819809721662668549555680, 4.25180349335114163320145476191, 4.35231551415535401904748027263, 4.39047882260428115452575234933, 4.68418667432380258451833795593, 4.71017628835552676561306985158, 4.72180629822626357837871862928, 4.91981947250966653096374003001, 5.22075209533499278869260106210, 5.42881904571923356394990334317

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

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