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Properties

Label 16-363e8-1.1-c2e8-0-0
Degree $16$
Conductor $3.015\times 10^{20}$
Sign $1$
Analytic cond. $9.16080\times 10^{7}$
Root an. cond. $3.14500$
Motivic weight $2$
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  − 5·3-s − 8·4-s + 9·9-s + 40·12-s + 16·16-s + 49·25-s − 74·31-s − 72·36-s − 50·37-s − 80·48-s + 98·49-s − 280·67-s − 245·75-s + 370·93-s − 190·97-s − 392·100-s + 380·103-s + 250·111-s + 592·124-s + 127-s + 131-s + 137-s + 139-s + 144·144-s − 490·147-s + 400·148-s + 149-s + ⋯
L(s)  = 1  − 5/3·3-s − 2·4-s + 9-s + 10/3·12-s + 16-s + 1.95·25-s − 2.38·31-s − 2·36-s − 1.35·37-s − 5/3·48-s + 2·49-s − 4.17·67-s − 3.26·75-s + 3.97·93-s − 1.95·97-s − 3.91·100-s + 3.68·103-s + 2.25·111-s + 4.77·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 144-s − 3.33·147-s + 2.70·148-s + 0.00671·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(9.16080\times 10^{7}\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 11^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01499007702\)
\(L(\frac12)\) \(\approx\) \(0.01499007702\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5 T + 16 T^{2} + 35 T^{3} + 31 T^{4} + 35 p^{2} T^{5} + 16 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} \)
11 \( 1 \)
good2 \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
5 \( ( 1 - T - 24 T^{2} + 49 T^{3} + 551 T^{4} + 49 p^{2} T^{5} - 24 p^{4} T^{6} - p^{6} T^{7} + p^{8} T^{8} )( 1 + T - 24 T^{2} - 49 T^{3} + 551 T^{4} - 49 p^{2} T^{5} - 24 p^{4} T^{6} + p^{6} T^{7} + p^{8} T^{8} ) \)
7 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
19 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - 35 T + p^{2} T^{2} )^{4}( 1 + 35 T + p^{2} T^{2} )^{4} \)
29 \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
31 \( ( 1 + 37 T + 408 T^{2} - 20461 T^{3} - 1149145 T^{4} - 20461 p^{2} T^{5} + 408 p^{4} T^{6} + 37 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 25 T - 744 T^{2} - 52825 T^{3} - 302089 T^{4} - 52825 p^{2} T^{5} - 744 p^{4} T^{6} + 25 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
43 \( ( 1 + p^{2} T^{2} )^{8} \)
47 \( ( 1 - 50 T + 291 T^{2} + 95900 T^{3} - 5437819 T^{4} + 95900 p^{2} T^{5} + 291 p^{4} T^{6} - 50 p^{6} T^{7} + p^{8} T^{8} )( 1 + 50 T + 291 T^{2} - 95900 T^{3} - 5437819 T^{4} - 95900 p^{2} T^{5} + 291 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} ) \)
53 \( ( 1 - 70 T + 2091 T^{2} + 50260 T^{3} - 9391819 T^{4} + 50260 p^{2} T^{5} + 2091 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} )( 1 + 70 T + 2091 T^{2} - 50260 T^{3} - 9391819 T^{4} - 50260 p^{2} T^{5} + 2091 p^{4} T^{6} + 70 p^{6} T^{7} + p^{8} T^{8} ) \)
59 \( ( 1 - 107 T + 7968 T^{2} - 480109 T^{3} + 23635055 T^{4} - 480109 p^{2} T^{5} + 7968 p^{4} T^{6} - 107 p^{6} T^{7} + p^{8} T^{8} )( 1 + 107 T + 7968 T^{2} + 480109 T^{3} + 23635055 T^{4} + 480109 p^{2} T^{5} + 7968 p^{4} T^{6} + 107 p^{6} T^{7} + p^{8} T^{8} ) \)
61 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 35 T + p^{2} T^{2} )^{8} \)
71 \( ( 1 - 133 T + 12648 T^{2} - 1011731 T^{3} + 70801655 T^{4} - 1011731 p^{2} T^{5} + 12648 p^{4} T^{6} - 133 p^{6} T^{7} + p^{8} T^{8} )( 1 + 133 T + 12648 T^{2} + 1011731 T^{3} + 70801655 T^{4} + 1011731 p^{2} T^{5} + 12648 p^{4} T^{6} + 133 p^{6} T^{7} + p^{8} T^{8} ) \)
73 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} - p^{6} T^{6} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2}( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
89 \( ( 1 - 97 T + p^{2} T^{2} )^{4}( 1 + 97 T + p^{2} T^{2} )^{4} \)
97 \( ( 1 + 95 T - 384 T^{2} - 930335 T^{3} - 84768769 T^{4} - 930335 p^{2} T^{5} - 384 p^{4} T^{6} + 95 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−4.86062257783906000068136296546, −4.79713485758522555622101550779, −4.51929791049235712820210233850, −4.40325368742249465968971933908, −4.30478680018051011428142730455, −4.21165340113189725491850466667, −4.10183051100646912883092893103, −3.80467210946420584630126957450, −3.73974588195325459230189200692, −3.36600568746237269420866389190, −3.34992038109763052053836456393, −3.15682398245965353697259572245, −3.12560631252409283237293311683, −2.76295687198298987409464097500, −2.65389595159377747391350088693, −2.30984882892034941942058075176, −2.11901223082422016696535796897, −1.98663282712181030875322580249, −1.55675684882983732842631525240, −1.51474209413340332609270905183, −1.27222538950813838168106715699, −0.77582668421686330427242968912, −0.72967833183705970713979538950, −0.35867406250259902248902031471, −0.03510110416635932729833286861, 0.03510110416635932729833286861, 0.35867406250259902248902031471, 0.72967833183705970713979538950, 0.77582668421686330427242968912, 1.27222538950813838168106715699, 1.51474209413340332609270905183, 1.55675684882983732842631525240, 1.98663282712181030875322580249, 2.11901223082422016696535796897, 2.30984882892034941942058075176, 2.65389595159377747391350088693, 2.76295687198298987409464097500, 3.12560631252409283237293311683, 3.15682398245965353697259572245, 3.34992038109763052053836456393, 3.36600568746237269420866389190, 3.73974588195325459230189200692, 3.80467210946420584630126957450, 4.10183051100646912883092893103, 4.21165340113189725491850466667, 4.30478680018051011428142730455, 4.40325368742249465968971933908, 4.51929791049235712820210233850, 4.79713485758522555622101550779, 4.86062257783906000068136296546

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

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