Properties

Label 16-882e8-1.1-c2e8-0-7
Degree $16$
Conductor $3.662\times 10^{23}$
Sign $1$
Analytic cond. $1.11283\times 10^{11}$
Root an. cond. $4.90232$
Motivic weight $2$
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  + 4·4-s − 64·13-s + 4·16-s − 80·19-s + 16·25-s − 16·31-s − 152·37-s + 160·43-s − 256·52-s − 232·61-s − 16·64-s + 192·67-s + 96·73-s − 320·76-s − 304·79-s − 576·97-s + 64·100-s − 272·103-s + 288·109-s − 20·121-s − 64·124-s + 127-s + 131-s + 137-s + 139-s − 608·148-s + 149-s + ⋯
L(s)  = 1  + 4-s − 4.92·13-s + 1/4·16-s − 4.21·19-s + 0.639·25-s − 0.516·31-s − 4.10·37-s + 3.72·43-s − 4.92·52-s − 3.80·61-s − 1/4·64-s + 2.86·67-s + 1.31·73-s − 4.21·76-s − 3.84·79-s − 5.93·97-s + 0.639·100-s − 2.64·103-s + 2.64·109-s − 0.165·121-s − 0.516·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 4.10·148-s + 0.00671·149-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.11283\times 10^{11}\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6920122990\)
\(L(\frac12)\) \(\approx\) \(0.6920122990\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 16 T^{2} - 122 p T^{4} + 6144 T^{6} + 199331 T^{8} + 6144 p^{4} T^{10} - 122 p^{9} T^{12} - 16 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 + 20 T^{2} - 21814 T^{4} - 141360 T^{6} + 273246515 T^{8} - 141360 p^{4} T^{10} - 21814 p^{8} T^{12} + 20 p^{12} T^{14} + p^{16} T^{16} \)
13 \( ( 1 + 16 T + 290 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
17 \( 1 + 368 T^{2} + 10238 T^{4} - 15403008 T^{6} - 2614970749 T^{8} - 15403008 p^{4} T^{10} + 10238 p^{8} T^{12} + 368 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 20 T + 39 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
23 \( 1 + 1652 T^{2} + 1494314 T^{4} + 1115278416 T^{6} + 682680166355 T^{8} + 1115278416 p^{4} T^{10} + 1494314 p^{8} T^{12} + 1652 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 - 1472 T^{2} + 1309346 T^{4} - 1472 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 8 T - 46 p T^{2} - 3456 T^{3} + 1235075 T^{4} - 3456 p^{2} T^{5} - 46 p^{5} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 38 T + 75 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 2800 T^{2} + 4672194 T^{4} - 2800 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 40 T + 66 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( ( 1 + 4130 T^{2} + 12177219 T^{4} + 4130 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( 1 - 4928 T^{2} + 3013534 T^{4} - 27058110464 T^{6} + 198601620377635 T^{8} - 27058110464 p^{4} T^{10} + 3013534 p^{8} T^{12} - 4928 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 + 10532 T^{2} + 61824746 T^{4} + 261862971792 T^{6} + 919716139941299 T^{8} + 261862971792 p^{4} T^{10} + 61824746 p^{8} T^{12} + 10532 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 116 T + 4442 T^{2} + 182352 T^{3} + 17336579 T^{4} + 182352 p^{2} T^{5} + 4442 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 - 96 T + 5102 T^{2} + 466944 T^{3} - 44596749 T^{4} + 466944 p^{2} T^{5} + 5102 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 19700 T^{2} + 147838694 T^{4} - 19700 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 48 T - 6130 T^{2} + 106752 T^{3} + 30456099 T^{4} + 106752 p^{2} T^{5} - 6130 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 152 T + 6638 T^{2} + 605568 T^{3} + 87987011 T^{4} + 605568 p^{2} T^{5} + 6638 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 9380 T^{2} + 87552614 T^{4} - 9380 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( 1 + 12208 T^{2} + 80677054 T^{4} - 697397528576 T^{6} - 8614550382684605 T^{8} - 697397528576 p^{4} T^{10} + 80677054 p^{8} T^{12} + 12208 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 + 144 T + 10450 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−4.14497387440632183359201467748, −4.10410245511532683661150114663, −4.07861566079055489242156128988, −3.82273639246621460262055251201, −3.40322138443682018572326270499, −3.29408018365350200546128548804, −3.28256553196275058173791449152, −3.25206946935532415342913591027, −2.95794310918139727579492937882, −2.76970534792255958066048598894, −2.69853946963617875354784602318, −2.51136934897904747840970781117, −2.48487901465152878734149588800, −2.15464752480378188193828104304, −2.07773734357593596796719437579, −2.05422216718695377163991415260, −1.92855990831688585391888783216, −1.82227945275977823303065214917, −1.48089979587551990523158002969, −1.44114190484581666442502698679, −1.00578862342427082412136940712, −0.62089362759310779318732035936, −0.49413956933431186125404200601, −0.18564317765379741502903736136, −0.17978572353605777820556918976, 0.17978572353605777820556918976, 0.18564317765379741502903736136, 0.49413956933431186125404200601, 0.62089362759310779318732035936, 1.00578862342427082412136940712, 1.44114190484581666442502698679, 1.48089979587551990523158002969, 1.82227945275977823303065214917, 1.92855990831688585391888783216, 2.05422216718695377163991415260, 2.07773734357593596796719437579, 2.15464752480378188193828104304, 2.48487901465152878734149588800, 2.51136934897904747840970781117, 2.69853946963617875354784602318, 2.76970534792255958066048598894, 2.95794310918139727579492937882, 3.25206946935532415342913591027, 3.28256553196275058173791449152, 3.29408018365350200546128548804, 3.40322138443682018572326270499, 3.82273639246621460262055251201, 4.07861566079055489242156128988, 4.10410245511532683661150114663, 4.14497387440632183359201467748

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

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