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Properties

Label 8-20e8-1.1-c5e4-0-1
Degree 88
Conductor 2560000000025600000000
Sign 11
Analytic cond. 1.69387×1071.69387\times 10^{7}
Root an. cond. 8.009588.00958
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 46·9-s + 120·11-s + 4.18e3·19-s − 7.10e3·29-s + 1.77e4·31-s − 2.48e4·41-s + 3.69e4·49-s − 7.33e4·59-s + 3.84e4·61-s − 5.47e3·71-s − 3.23e4·79-s + 4.70e4·81-s + 9.46e4·89-s − 5.52e3·99-s + 2.79e5·101-s + 1.93e5·109-s − 4.51e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.90e5·169-s + ⋯
L(s)  = 1  − 0.189·9-s + 0.299·11-s + 2.65·19-s − 1.56·29-s + 3.32·31-s − 2.31·41-s + 2.20·49-s − 2.74·59-s + 1.32·61-s − 0.128·71-s − 0.583·79-s + 0.797·81-s + 1.26·89-s − 0.0566·99-s + 2.72·101-s + 1.56·109-s − 2.80·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.59·169-s + ⋯

Functional equation

Λ(s)=((216⋅58)s/2 ΓC(s)4 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((216⋅58)s/2 ΓC(s+5/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅582^{16} \cdot 5^{8}
Sign: 11
Analytic conductor: 1.69387×1071.69387\times 10^{7}
Root analytic conductor: 8.009588.00958
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅58, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{16} \cdot 5^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 0.96585755010.9658575501
L(12)L(\frac12) ≈\approx 0.96585755010.9658575501
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5 1 1
good3D4×C2D_4\times C_2 1+46T2−4997p2T4+46p10T6+p20T8 1 + 46 T^{2} - 4997 p^{2} T^{4} + 46 p^{10} T^{6} + p^{20} T^{8}
7D4×C2D_4\times C_2 1−36980T2+883272198T4−36980p10T6+p20T8 1 - 36980 T^{2} + 883272198 T^{4} - 36980 p^{10} T^{6} + p^{20} T^{8}
11D4D_{4} (1−60T+230977T2−60p5T3+p10T4)2 ( 1 - 60 T + 230977 T^{2} - 60 p^{5} T^{3} + p^{10} T^{4} )^{2}
13D4×C2D_4\times C_2 1−590804T2+163581027702T4−590804p10T6+p20T8 1 - 590804 T^{2} + 163581027702 T^{4} - 590804 p^{10} T^{6} + p^{20} T^{8}
17D4×C2D_4\times C_2 1−1327586T2+3973871730147T4−1327586p10T6+p20T8 1 - 1327586 T^{2} + 3973871730147 T^{4} - 1327586 p^{10} T^{6} + p^{20} T^{8}
19D4D_{4} (1−2092T+5218089T2−2092p5T3+p10T4)2 ( 1 - 2092 T + 5218089 T^{2} - 2092 p^{5} T^{3} + p^{10} T^{4} )^{2}
23D4×C2D_4\times C_2 1−10160180T2+108548175292998T4−10160180p10T6+p20T8 1 - 10160180 T^{2} + 108548175292998 T^{4} - 10160180 p^{10} T^{6} + p^{20} T^{8}
29D4D_{4} (1+3552T+20618074T2+3552p5T3+p10T4)2 ( 1 + 3552 T + 20618074 T^{2} + 3552 p^{5} T^{3} + p^{10} T^{4} )^{2}
31D4D_{4} (1−8888T+67804938T2−8888p5T3+p10T4)2 ( 1 - 8888 T + 67804938 T^{2} - 8888 p^{5} T^{3} + p^{10} T^{4} )^{2}
37D4×C2D_4\times C_2 1−33594380T2−2634705186058602T4−33594380p10T6+p20T8 1 - 33594380 T^{2} - 2634705186058602 T^{4} - 33594380 p^{10} T^{6} + p^{20} T^{8}
41D4D_{4} (1+12438T+217381963T2+12438p5T3+p10T4)2 ( 1 + 12438 T + 217381963 T^{2} + 12438 p^{5} T^{3} + p^{10} T^{4} )^{2}
43D4×C2D_4\times C_2 1−540244172T2+116157205788519894T4−540244172p10T6+p20T8 1 - 540244172 T^{2} + 116157205788519894 T^{4} - 540244172 p^{10} T^{6} + p^{20} T^{8}
47D4×C2D_4\times C_2 1−902407196T2+308772689280765702T4−902407196p10T6+p20T8 1 - 902407196 T^{2} + 308772689280765702 T^{4} - 902407196 p^{10} T^{6} + p^{20} T^{8}
53D4×C2D_4\times C_2 1−1189716620T2+656395125486896598T4−1189716620p10T6+p20T8 1 - 1189716620 T^{2} + 656395125486896598 T^{4} - 1189716620 p^{10} T^{6} + p^{20} T^{8}
59D4D_{4} (1+36696T+1234961302T2+36696p5T3+p10T4)2 ( 1 + 36696 T + 1234961302 T^{2} + 36696 p^{5} T^{3} + p^{10} T^{4} )^{2}
61D4D_{4} (1−19204T+627029406T2−19204p5T3+p10T4)2 ( 1 - 19204 T + 627029406 T^{2} - 19204 p^{5} T^{3} + p^{10} T^{4} )^{2}
67D4×C2D_4\times C_2 1−974988050T2+2516736182389071123T4−974988050p10T6+p20T8 1 - 974988050 T^{2} + 2516736182389071123 T^{4} - 974988050 p^{10} T^{6} + p^{20} T^{8}
71D4D_{4} (1+2736T+2006886526T2+2736p5T3+p10T4)2 ( 1 + 2736 T + 2006886526 T^{2} + 2736 p^{5} T^{3} + p^{10} T^{4} )^{2}
73D4×C2D_4\times C_2 1−8204978114T2+25425197509477462947T4−8204978114p10T6+p20T8 1 - 8204978114 T^{2} + 25425197509477462947 T^{4} - 8204978114 p^{10} T^{6} + p^{20} T^{8}
79D4D_{4} (1+16184T+6157384362T2+16184p5T3+p10T4)2 ( 1 + 16184 T + 6157384362 T^{2} + 16184 p^{5} T^{3} + p^{10} T^{4} )^{2}
83D4×C2D_4\times C_2 1−7195965410T2+40258768525685533923T4−7195965410p10T6+p20T8 1 - 7195965410 T^{2} + 40258768525685533923 T^{4} - 7195965410 p^{10} T^{6} + p^{20} T^{8}
89D4D_{4} (1−47322T−1006825781T2−47322p5T3+p10T4)2 ( 1 - 47322 T - 1006825781 T^{2} - 47322 p^{5} T^{3} + p^{10} T^{4} )^{2}
97D4×C2D_4\times C_2 1−28221621500T2+ 1 - 28221621500 T^{2} + 34 ⁣⋯ ⁣9834\!\cdots\!98T4−28221621500p10T6+p20T8 T^{4} - 28221621500 p^{10} T^{6} + p^{20} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.40398713280085432629081152280, −7.05730360635068593050478553473, −6.79433910624980744498318527138, −6.77778309001600587298953628413, −6.11380078456916611540432374790, −6.09163690323236799518726144253, −5.76935252129977773956774117662, −5.57176660603347047522385082644, −5.40308105845601135433634332596, −4.80883715713246131522954960375, −4.67583728887146231511670662941, −4.61091947168498199107241131299, −4.33896773780196025697692108380, −3.59842327603511593353697072462, −3.45486830732101766997968249577, −3.34280064963195514457218340424, −3.18188677791884735790007258408, −2.59144080785299611370255899838, −2.36029141612556492280851388664, −1.96363013665438110520346777903, −1.69839223623848856564687725124, −1.02096484997895471496647747888, −0.974126018263220829122564955966, −0.826187811440256196307099789058, −0.10543515571588512049773446391, 0.10543515571588512049773446391, 0.826187811440256196307099789058, 0.974126018263220829122564955966, 1.02096484997895471496647747888, 1.69839223623848856564687725124, 1.96363013665438110520346777903, 2.36029141612556492280851388664, 2.59144080785299611370255899838, 3.18188677791884735790007258408, 3.34280064963195514457218340424, 3.45486830732101766997968249577, 3.59842327603511593353697072462, 4.33896773780196025697692108380, 4.61091947168498199107241131299, 4.67583728887146231511670662941, 4.80883715713246131522954960375, 5.40308105845601135433634332596, 5.57176660603347047522385082644, 5.76935252129977773956774117662, 6.09163690323236799518726144253, 6.11380078456916611540432374790, 6.77778309001600587298953628413, 6.79433910624980744498318527138, 7.05730360635068593050478553473, 7.40398713280085432629081152280

Graph of the ZZ-function along the critical line