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Properties

Label 8-525e4-1.1-c3e4-0-10
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 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 12·3-s + 10·4-s + 72·6-s + 28·7-s − 3·8-s + 90·9-s + 57·11-s − 120·12-s − 43·13-s − 168·14-s + 54·16-s − 99·17-s − 540·18-s − 12·19-s − 336·21-s − 342·22-s − 156·23-s + 36·24-s + 258·26-s − 540·27-s + 280·28-s + 378·29-s − 93·31-s − 240·32-s − 684·33-s + 594·34-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 5/4·4-s + 4.89·6-s + 1.51·7-s − 0.132·8-s + 10/3·9-s + 1.56·11-s − 2.88·12-s − 0.917·13-s − 3.20·14-s + 0.843·16-s − 1.41·17-s − 7.07·18-s − 0.144·19-s − 3.49·21-s − 3.31·22-s − 1.41·23-s + 0.306·24-s + 1.94·26-s − 3.84·27-s + 1.88·28-s + 2.42·29-s − 0.538·31-s − 1.32·32-s − 3.60·33-s + 2.99·34-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: 44
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) == 00
L(12)L(\frac12) == 00
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)
bad3C1C_1 (1+pT)4 ( 1 + p T )^{4}
5 1 1
7C1C_1 (1−pT)4 ( 1 - p T )^{4}
good2C2≀C2≀C2C_2 \wr C_2\wr C_2 1+3pT+13pT2+99T3+149pT4+99p3T5+13p7T6+3p10T7+p12T8 1 + 3 p T + 13 p T^{2} + 99 T^{3} + 149 p T^{4} + 99 p^{3} T^{5} + 13 p^{7} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8}
11C2≀C2≀C2C_2 \wr C_2\wr C_2 1−57T+4001T2−182772T3+7206874T4−182772p3T5+4001p6T6−57p9T7+p12T8 1 - 57 T + 4001 T^{2} - 182772 T^{3} + 7206874 T^{4} - 182772 p^{3} T^{5} + 4001 p^{6} T^{6} - 57 p^{9} T^{7} + p^{12} T^{8}
13C2≀C2≀C2C_2 \wr C_2\wr C_2 1+43T+4464T2+221093T3+13201958T4+221093p3T5+4464p6T6+43p9T7+p12T8 1 + 43 T + 4464 T^{2} + 221093 T^{3} + 13201958 T^{4} + 221093 p^{3} T^{5} + 4464 p^{6} T^{6} + 43 p^{9} T^{7} + p^{12} T^{8}
17C2≀C2≀C2C_2 \wr C_2\wr C_2 1+99T+13790T2+926229T3+90687946T4+926229p3T5+13790p6T6+99p9T7+p12T8 1 + 99 T + 13790 T^{2} + 926229 T^{3} + 90687946 T^{4} + 926229 p^{3} T^{5} + 13790 p^{6} T^{6} + 99 p^{9} T^{7} + p^{12} T^{8}
19C2≀C2≀C2C_2 \wr C_2\wr C_2 1+12T−10212T2+154020T3+95206646T4+154020p3T5−10212p6T6+12p9T7+p12T8 1 + 12 T - 10212 T^{2} + 154020 T^{3} + 95206646 T^{4} + 154020 p^{3} T^{5} - 10212 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8}
23C2≀C2≀C2C_2 \wr C_2\wr C_2 1+156T+28082T2+3613896T3+479553835T4+3613896p3T5+28082p6T6+156p9T7+p12T8 1 + 156 T + 28082 T^{2} + 3613896 T^{3} + 479553835 T^{4} + 3613896 p^{3} T^{5} + 28082 p^{6} T^{6} + 156 p^{9} T^{7} + p^{12} T^{8}
29C2≀C2≀C2C_2 \wr C_2\wr C_2 1−378T+143760T2−1025172pT3+5852097929T4−1025172p4T5+143760p6T6−378p9T7+p12T8 1 - 378 T + 143760 T^{2} - 1025172 p T^{3} + 5852097929 T^{4} - 1025172 p^{4} T^{5} + 143760 p^{6} T^{6} - 378 p^{9} T^{7} + p^{12} T^{8}
31C2≀C2≀C2C_2 \wr C_2\wr C_2 1+3pT+16330T2−936891T3+385099458T4−936891p3T5+16330p6T6+3p10T7+p12T8 1 + 3 p T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 936891 p^{3} T^{5} + 16330 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8}
37C2≀C2≀C2C_2 \wr C_2\wr C_2 1+81T+107185T2+469746T3+5937897606T4+469746p3T5+107185p6T6+81p9T7+p12T8 1 + 81 T + 107185 T^{2} + 469746 T^{3} + 5937897606 T^{4} + 469746 p^{3} T^{5} + 107185 p^{6} T^{6} + 81 p^{9} T^{7} + p^{12} T^{8}
41C2≀C2≀C2C_2 \wr C_2\wr C_2 1+465T+105716T2+566235T3−2050285754T4+566235p3T5+105716p6T6+465p9T7+p12T8 1 + 465 T + 105716 T^{2} + 566235 T^{3} - 2050285754 T^{4} + 566235 p^{3} T^{5} + 105716 p^{6} T^{6} + 465 p^{9} T^{7} + p^{12} T^{8}
43C2≀C2≀C2C_2 \wr C_2\wr C_2 1−64T+64662T2+2492176T3+5477627255T4+2492176p3T5+64662p6T6−64p9T7+p12T8 1 - 64 T + 64662 T^{2} + 2492176 T^{3} + 5477627255 T^{4} + 2492176 p^{3} T^{5} + 64662 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8}
47C2≀C2≀C2C_2 \wr C_2\wr C_2 1+744T+588276T2+247094976T3+101020317878T4+247094976p3T5+588276p6T6+744p9T7+p12T8 1 + 744 T + 588276 T^{2} + 247094976 T^{3} + 101020317878 T^{4} + 247094976 p^{3} T^{5} + 588276 p^{6} T^{6} + 744 p^{9} T^{7} + p^{12} T^{8}
53C2≀C2≀C2C_2 \wr C_2\wr C_2 1+729T+333446T2+137056347T3+63585804370T4+137056347p3T5+333446p6T6+729p9T7+p12T8 1 + 729 T + 333446 T^{2} + 137056347 T^{3} + 63585804370 T^{4} + 137056347 p^{3} T^{5} + 333446 p^{6} T^{6} + 729 p^{9} T^{7} + p^{12} T^{8}
59C2≀C2≀C2C_2 \wr C_2\wr C_2 1−231T+701280T2−148812447T3+204019205798T4−148812447p3T5+701280p6T6−231p9T7+p12T8 1 - 231 T + 701280 T^{2} - 148812447 T^{3} + 204019205798 T^{4} - 148812447 p^{3} T^{5} + 701280 p^{6} T^{6} - 231 p^{9} T^{7} + p^{12} T^{8}
61C2≀C2≀C2C_2 \wr C_2\wr C_2 1+1353T+1374808T2+931878711T3+511309944510T4+931878711p3T5+1374808p6T6+1353p9T7+p12T8 1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 931878711 p^{3} T^{5} + 1374808 p^{6} T^{6} + 1353 p^{9} T^{7} + p^{12} T^{8}
67C2≀C2≀C2C_2 \wr C_2\wr C_2 1+1487T+1603371T2+1080212116T3+674138302124T4+1080212116p3T5+1603371p6T6+1487p9T7+p12T8 1 + 1487 T + 1603371 T^{2} + 1080212116 T^{3} + 674138302124 T^{4} + 1080212116 p^{3} T^{5} + 1603371 p^{6} T^{6} + 1487 p^{9} T^{7} + p^{12} T^{8}
71C2≀C2≀C2C_2 \wr C_2\wr C_2 1+1725T+1434819T2+706812600T3+347044577276T4+706812600p3T5+1434819p6T6+1725p9T7+p12T8 1 + 1725 T + 1434819 T^{2} + 706812600 T^{3} + 347044577276 T^{4} + 706812600 p^{3} T^{5} + 1434819 p^{6} T^{6} + 1725 p^{9} T^{7} + p^{12} T^{8}
73C2≀C2≀C2C_2 \wr C_2\wr C_2 1+512T+1030980T2+484092568T3+566560915526T4+484092568p3T5+1030980p6T6+512p9T7+p12T8 1 + 512 T + 1030980 T^{2} + 484092568 T^{3} + 566560915526 T^{4} + 484092568 p^{3} T^{5} + 1030980 p^{6} T^{6} + 512 p^{9} T^{7} + p^{12} T^{8}
79C2≀C2≀C2C_2 \wr C_2\wr C_2 1−1629T+1989837T2−1837769472T3+1504809477470T4−1837769472p3T5+1989837p6T6−1629p9T7+p12T8 1 - 1629 T + 1989837 T^{2} - 1837769472 T^{3} + 1504809477470 T^{4} - 1837769472 p^{3} T^{5} + 1989837 p^{6} T^{6} - 1629 p^{9} T^{7} + p^{12} T^{8}
83C2≀C2≀C2C_2 \wr C_2\wr C_2 1+321T+661434T2+144132717T3+79870144034T4+144132717p3T5+661434p6T6+321p9T7+p12T8 1 + 321 T + 661434 T^{2} + 144132717 T^{3} + 79870144034 T^{4} + 144132717 p^{3} T^{5} + 661434 p^{6} T^{6} + 321 p^{9} T^{7} + p^{12} T^{8}
89C2≀C2≀C2C_2 \wr C_2\wr C_2 1+978T+526228T2−159100050T3−360762420714T4−159100050p3T5+526228p6T6+978p9T7+p12T8 1 + 978 T + 526228 T^{2} - 159100050 T^{3} - 360762420714 T^{4} - 159100050 p^{3} T^{5} + 526228 p^{6} T^{6} + 978 p^{9} T^{7} + p^{12} T^{8}
97C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2616T+5183100T2+6945540936T3+7789652954246T4+6945540936p3T5+5183100p6T6+2616p9T7+p12T8 1 + 2616 T + 5183100 T^{2} + 6945540936 T^{3} + 7789652954246 T^{4} + 6945540936 p^{3} T^{5} + 5183100 p^{6} T^{6} + 2616 p^{9} T^{7} + 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

−8.054825066716703602517723721220, −7.55249737699634288167419862459, −7.47969872058425052966787871745, −7.35377986734571396282747206603, −6.73855213728489332593083648611, −6.72342074589394933126000025308, −6.48153370464659226231671852917, −6.33005565602186449911696913972, −5.99013560424541777607107900558, −5.89167459054678131855702884977, −5.40041748608843241685662499846, −5.07595406855566013906505098822, −5.03925054197260239810462418554, −4.61645302411216913350267400647, −4.42183490475584585989613830284, −4.35283915284952835623341398061, −4.08078949704708557499859074312, −3.43651309869142469404583537475, −3.26744080394513471324152703770, −2.65294018966755217437324764249, −2.31094728544811242299797755097, −1.71802940239297374289343109910, −1.37009884543551029158262571366, −1.27583858606173836779918367966, −1.21046594203842898808588622703, 0, 0, 0, 0, 1.21046594203842898808588622703, 1.27583858606173836779918367966, 1.37009884543551029158262571366, 1.71802940239297374289343109910, 2.31094728544811242299797755097, 2.65294018966755217437324764249, 3.26744080394513471324152703770, 3.43651309869142469404583537475, 4.08078949704708557499859074312, 4.35283915284952835623341398061, 4.42183490475584585989613830284, 4.61645302411216913350267400647, 5.03925054197260239810462418554, 5.07595406855566013906505098822, 5.40041748608843241685662499846, 5.89167459054678131855702884977, 5.99013560424541777607107900558, 6.33005565602186449911696913972, 6.48153370464659226231671852917, 6.72342074589394933126000025308, 6.73855213728489332593083648611, 7.35377986734571396282747206603, 7.47969872058425052966787871745, 7.55249737699634288167419862459, 8.054825066716703602517723721220

Graph of the ZZ-function along the critical line