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Properties

Label 8-20e8-1.1-c5e4-0-2
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  + 458·9-s + 400·11-s − 1.68e3·19-s + 9.36e3·29-s + 1.00e4·31-s − 1.06e4·41-s + 6.52e4·49-s − 1.63e5·59-s − 9.38e4·61-s − 1.48e4·71-s − 2.16e5·79-s + 5.46e4·81-s − 1.41e5·89-s + 1.83e5·99-s − 3.02e5·101-s − 2.82e5·109-s − 3.31e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.61e3·169-s + ⋯
L(s)  = 1  + 1.88·9-s + 0.996·11-s − 1.06·19-s + 2.06·29-s + 1.87·31-s − 0.991·41-s + 3.88·49-s − 6.11·59-s − 3.22·61-s − 0.350·71-s − 3.89·79-s + 0.925·81-s − 1.89·89-s + 1.87·99-s − 2.94·101-s − 2.28·109-s − 2.05·121-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 + 0.0178·169-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 1.1073076681.107307668
L(12)L(\frac12) ≈\approx 1.1073076681.107307668
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 1−458T2+1915p4T4−458p10T6+p20T8 1 - 458 T^{2} + 1915 p^{4} T^{4} - 458 p^{10} T^{6} + p^{20} T^{8}
7D4×C2D_4\times C_2 1−1332p2T2+1629866758T4−1332p12T6+p20T8 1 - 1332 p^{2} T^{2} + 1629866758 T^{4} - 1332 p^{12} T^{6} + p^{20} T^{8}
11D4D_{4} (1−200T+225821T2−200p5T3+p10T4)2 ( 1 - 200 T + 225821 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4} )^{2}
13D4×C2D_4\times C_2 1−6612T2+47343141238T4−6612p10T6+p20T8 1 - 6612 T^{2} + 47343141238 T^{4} - 6612 p^{10} T^{6} + p^{20} T^{8}
17D4×C2D_4\times C_2 1−5517394T2+11637628825043T4−5517394p10T6+p20T8 1 - 5517394 T^{2} + 11637628825043 T^{4} - 5517394 p^{10} T^{6} + p^{20} T^{8}
19D4D_{4} (1+840T+4289677T2+840p5T3+p10T4)2 ( 1 + 840 T + 4289677 T^{2} + 840 p^{5} T^{3} + p^{10} T^{4} )^{2}
23D4×C2D_4\times C_2 1−19211092T2+166300343917958T4−19211092p10T6+p20T8 1 - 19211092 T^{2} + 166300343917958 T^{4} - 19211092 p^{10} T^{6} + p^{20} T^{8}
29D4D_{4} (1−4680T+1421346pT2−4680p5T3+p10T4)2 ( 1 - 4680 T + 1421346 p T^{2} - 4680 p^{5} T^{3} + p^{10} T^{4} )^{2}
31D4D_{4} (1−5008T+33327162T2−5008p5T3+p10T4)2 ( 1 - 5008 T + 33327162 T^{2} - 5008 p^{5} T^{3} + p^{10} T^{4} )^{2}
37D4×C2D_4\times C_2 1−86339436T2+2659590797035222T4−86339436p10T6+p20T8 1 - 86339436 T^{2} + 2659590797035222 T^{4} - 86339436 p^{10} T^{6} + p^{20} T^{8}
41D4D_{4} (1+5334T+27686155T2+5334p5T3+p10T4)2 ( 1 + 5334 T + 27686155 T^{2} + 5334 p^{5} T^{3} + p^{10} T^{4} )^{2}
43D4×C2D_4\times C_2 1−29960652T2+43433374484607958T4−29960652p10T6+p20T8 1 - 29960652 T^{2} + 43433374484607958 T^{4} - 29960652 p^{10} T^{6} + p^{20} T^{8}
47C22C_2^2 (1−288752718T2+p10T4)2 ( 1 - 288752718 T^{2} + p^{10} T^{4} )^{2}
53D4×C2D_4\times C_2 1−514633100T2+347542581493770198T4−514633100p10T6+p20T8 1 - 514633100 T^{2} + 347542581493770198 T^{4} - 514633100 p^{10} T^{6} + p^{20} T^{8}
59D4D_{4} (1+81776T+3059970646T2+81776p5T3+p10T4)2 ( 1 + 81776 T + 3059970646 T^{2} + 81776 p^{5} T^{3} + p^{10} T^{4} )^{2}
61D4D_{4} (1+46932T+2239379182T2+46932p5T3+p10T4)2 ( 1 + 46932 T + 2239379182 T^{2} + 46932 p^{5} T^{3} + p^{10} T^{4} )^{2}
67D4×C2D_4\times C_2 1−2982943418T2+5751121538625041883T4−2982943418p10T6+p20T8 1 - 2982943418 T^{2} + 5751121538625041883 T^{4} - 2982943418 p^{10} T^{6} + p^{20} T^{8}
71D4D_{4} (1+7448T+3593807902T2+7448p5T3+p10T4)2 ( 1 + 7448 T + 3593807902 T^{2} + 7448 p^{5} T^{3} + p^{10} T^{4} )^{2}
73D4×C2D_4\times C_2 1−2368673842T2+9983110585747236243T4−2368673842p10T6+p20T8 1 - 2368673842 T^{2} + 9983110585747236243 T^{4} - 2368673842 p^{10} T^{6} + p^{20} T^{8}
79D4D_{4} (1+108104T+6816153098T2+108104p5T3+p10T4)2 ( 1 + 108104 T + 6816153098 T^{2} + 108104 p^{5} T^{3} + p^{10} T^{4} )^{2}
83D4×C2D_4\times C_2 1−8274508042T2+45513777799710942443T4−8274508042p10T6+p20T8 1 - 8274508042 T^{2} + 45513777799710942443 T^{4} - 8274508042 p^{10} T^{6} + p^{20} T^{8}
89D4D_{4} (1+70990T+6345035107T2+70990p5T3+p10T4)2 ( 1 + 70990 T + 6345035107 T^{2} + 70990 p^{5} T^{3} + p^{10} T^{4} )^{2}
97D4×C2D_4\times C_2 1−22699126076T2+ 1 - 22699126076 T^{2} + 24 ⁣⋯ ⁣4224\!\cdots\!42T4−22699126076p10T6+p20T8 T^{4} - 22699126076 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

−7.22171930189438933714770280347, −7.18149782651507435171991932914, −6.92056930323724152326141421754, −6.35190325457508689774859263014, −6.24870478256814141059083090402, −6.24655831961947912347772023758, −6.11243150780367246229444345206, −5.51126758930304394086993144781, −5.13284249925628783845851612813, −4.94753973206135619990763114506, −4.57507322350901849406156385268, −4.28639793002614092324151207971, −4.27634963470377505033110473711, −3.96797720615073122359533515122, −3.89774485908660560905560629565, −2.99789916525418945741066963265, −2.92925878728820146068957485401, −2.72809883092505504913273257748, −2.54182571481102836277657297841, −1.64551414249457844587245147104, −1.39210243294453318986926915077, −1.36512564553566985843762672332, −1.36185935200599537084277574108, −0.57078583167653498487124373135, −0.11960983469108833787289390937, 0.11960983469108833787289390937, 0.57078583167653498487124373135, 1.36185935200599537084277574108, 1.36512564553566985843762672332, 1.39210243294453318986926915077, 1.64551414249457844587245147104, 2.54182571481102836277657297841, 2.72809883092505504913273257748, 2.92925878728820146068957485401, 2.99789916525418945741066963265, 3.89774485908660560905560629565, 3.96797720615073122359533515122, 4.27634963470377505033110473711, 4.28639793002614092324151207971, 4.57507322350901849406156385268, 4.94753973206135619990763114506, 5.13284249925628783845851612813, 5.51126758930304394086993144781, 6.11243150780367246229444345206, 6.24655831961947912347772023758, 6.24870478256814141059083090402, 6.35190325457508689774859263014, 6.92056930323724152326141421754, 7.18149782651507435171991932914, 7.22171930189438933714770280347

Graph of the ZZ-function along the critical line