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Properties

Label 8-525e4-1.1-c3e4-0-1
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 920664.920664.
Root an. cond. 5.565605.56560
Motivic weight 33
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  − 4-s − 18·9-s − 44·11-s − 111·16-s − 204·19-s + 392·29-s + 300·31-s + 18·36-s − 352·41-s + 44·44-s − 98·49-s + 1.68e3·59-s − 408·61-s + 159·64-s + 3.34e3·71-s + 204·76-s + 1.77e3·79-s + 243·81-s − 1.17e3·89-s + 792·99-s − 64·101-s + 608·109-s − 392·116-s − 3.98e3·121-s − 300·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/8·4-s − 2/3·9-s − 1.20·11-s − 1.73·16-s − 2.46·19-s + 2.51·29-s + 1.73·31-s + 1/12·36-s − 1.34·41-s + 0.150·44-s − 2/7·49-s + 3.72·59-s − 0.856·61-s + 0.310·64-s + 5.58·71-s + 0.307·76-s + 2.52·79-s + 1/3·81-s − 1.40·89-s + 0.804·99-s − 0.0630·101-s + 0.534·109-s − 0.313·116-s − 2.99·121-s − 0.217·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 920664.920664.
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.19233247720.1923324772
L(12)L(\frac12) ≈\approx 0.19233247720.1923324772
L(52)L(\frac{5}{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)
bad3C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
5 1 1
7C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
good2D4×C2D_4\times C_2 1+T2+7p4T4+p6T6+p12T8 1 + T^{2} + 7 p^{4} T^{4} + p^{6} T^{6} + p^{12} T^{8}
11D4D_{4} (1+2pT+2718T2+2p4T3+p6T4)2 ( 1 + 2 p T + 2718 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} )^{2}
13D4×C2D_4\times C_2 1−8416T2+27329422T4−8416p6T6+p12T8 1 - 8416 T^{2} + 27329422 T^{4} - 8416 p^{6} T^{6} + p^{12} T^{8}
17D4×C2D_4\times C_2 1−4604T2−2402618T4−4604p6T6+p12T8 1 - 4604 T^{2} - 2402618 T^{4} - 4604 p^{6} T^{6} + p^{12} T^{8}
19D4D_{4} (1+102T+13134T2+102p3T3+p6T4)2 ( 1 + 102 T + 13134 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} )^{2}
23D4×C2D_4\times C_2 1−1868T2−142455866T4−1868p6T6+p12T8 1 - 1868 T^{2} - 142455866 T^{4} - 1868 p^{6} T^{6} + p^{12} T^{8}
29D4D_{4} (1−196T+20942T2−196p3T3+p6T4)2 ( 1 - 196 T + 20942 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} )^{2}
31D4D_{4} (1−150T+36542T2−150p3T3+p6T4)2 ( 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} )^{2}
37D4×C2D_4\times C_2 1−155884T2+11012319222T4−155884p6T6+p12T8 1 - 155884 T^{2} + 11012319222 T^{4} - 155884 p^{6} T^{6} + p^{12} T^{8}
41D4D_{4} (1+176T−16914T2+176p3T3+p6T4)2 ( 1 + 176 T - 16914 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} )^{2}
43D4×C2D_4\times C_2 1−225580T2+23395199158T4−225580p6T6+p12T8 1 - 225580 T^{2} + 23395199158 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8}
47D4×C2D_4\times C_2 1−183612T2+18245588294T4−183612p6T6+p12T8 1 - 183612 T^{2} + 18245588294 T^{4} - 183612 p^{6} T^{6} + p^{12} T^{8}
53D4×C2D_4\times C_2 1−302000T2+54356941198T4−302000p6T6+p12T8 1 - 302000 T^{2} + 54356941198 T^{4} - 302000 p^{6} T^{6} + p^{12} T^{8}
59D4D_{4} (1−844T+474182T2−844p3T3+p6T4)2 ( 1 - 844 T + 474182 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} )^{2}
61D4D_{4} (1+204T+455006T2+204p3T3+p6T4)2 ( 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2}
67D4×C2D_4\times C_2 1−1064524T2+463499688022T4−1064524p6T6+p12T8 1 - 1064524 T^{2} + 463499688022 T^{4} - 1064524 p^{6} T^{6} + p^{12} T^{8}
71D4D_{4} (1−1670T+1384382T2−1670p3T3+p6T4)2 ( 1 - 1670 T + 1384382 T^{2} - 1670 p^{3} T^{3} + p^{6} T^{4} )^{2}
73D4×C2D_4\times C_2 1−155440T2+209914818238T4−155440p6T6+p12T8 1 - 155440 T^{2} + 209914818238 T^{4} - 155440 p^{6} T^{6} + p^{12} T^{8}
79D4D_{4} (1−888T+1007454T2−888p3T3+p6T4)2 ( 1 - 888 T + 1007454 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} )^{2}
83D4×C2D_4\times C_2 1−340236T2+29905619222T4−340236p6T6+p12T8 1 - 340236 T^{2} + 29905619222 T^{4} - 340236 p^{6} T^{6} + p^{12} T^{8}
89D4D_{4} (1+588T+1495334T2+588p3T3+p6T4)2 ( 1 + 588 T + 1495334 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} )^{2}
97D4×C2D_4\times C_2 1−310080T2+1253411633918T4−310080p6T6+p12T8 1 - 310080 T^{2} + 1253411633918 T^{4} - 310080 p^{6} T^{6} + p^{12} 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

−7.46468960638725468710410316764, −6.85758866460490217686195210065, −6.81884674080411514778326676708, −6.70563780339687115985473973071, −6.45658335329951133032491410051, −6.31714700248074349796356714708, −6.21815021159355175710758615853, −5.47646647078565872685525504207, −5.24789212667538642918090425330, −5.15095990825133038978404529576, −5.12339337988887692799664641898, −4.52290368930383097628961529034, −4.33430164799673217844166853276, −4.31358716731988401315353683963, −3.65464113277708188772499867254, −3.64905251990573294445280500922, −3.16997452289445418459853478361, −2.62292098925399939831972819075, −2.37717022641922166950124653938, −2.34224726254288258447271356778, −2.25852779927536283359697924654, −1.50147145185159353282173626839, −0.868343270078126560103328904762, −0.69665451496178968429292956999, −0.07812670829720184653250556573, 0.07812670829720184653250556573, 0.69665451496178968429292956999, 0.868343270078126560103328904762, 1.50147145185159353282173626839, 2.25852779927536283359697924654, 2.34224726254288258447271356778, 2.37717022641922166950124653938, 2.62292098925399939831972819075, 3.16997452289445418459853478361, 3.64905251990573294445280500922, 3.65464113277708188772499867254, 4.31358716731988401315353683963, 4.33430164799673217844166853276, 4.52290368930383097628961529034, 5.12339337988887692799664641898, 5.15095990825133038978404529576, 5.24789212667538642918090425330, 5.47646647078565872685525504207, 6.21815021159355175710758615853, 6.31714700248074349796356714708, 6.45658335329951133032491410051, 6.70563780339687115985473973071, 6.81884674080411514778326676708, 6.85758866460490217686195210065, 7.46468960638725468710410316764

Graph of the ZZ-function along the critical line