Properties

Label 8-294e4-1.1-c5e4-0-1
Degree 88
Conductor 74711820967471182096
Sign 11
Analytic cond. 4.94346×1064.94346\times 10^{6}
Root an. cond. 6.866796.86679
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  − 8·2-s − 18·3-s + 16·4-s − 53·5-s + 144·6-s + 128·8-s + 81·9-s + 424·10-s − 191·11-s − 288·12-s + 758·13-s + 954·15-s − 1.02e3·16-s − 340·17-s − 648·18-s − 1.76e3·19-s − 848·20-s + 1.52e3·22-s − 3.23e3·23-s − 2.30e3·24-s + 4.55e3·25-s − 6.06e3·26-s + 1.45e3·27-s + 8.91e3·29-s − 7.63e3·30-s + 1.99e3·31-s + 2.04e3·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.948·5-s + 1.63·6-s + 0.707·8-s + 1/3·9-s + 1.34·10-s − 0.475·11-s − 0.577·12-s + 1.24·13-s + 1.09·15-s − 16-s − 0.285·17-s − 0.471·18-s − 1.12·19-s − 0.474·20-s + 0.673·22-s − 1.27·23-s − 0.816·24-s + 1.45·25-s − 1.75·26-s + 0.384·27-s + 1.96·29-s − 1.54·30-s + 0.372·31-s + 0.353·32-s + ⋯

Functional equation

Λ(s)=((243478)s/2ΓC(s)4L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((243478)s/2ΓC(s+5/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 2434782^{4} \cdot 3^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 4.94346×1064.94346\times 10^{6}
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 243478, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) \approx 0.080901343320.08090134332
L(12)L(\frac12) \approx 0.080901343320.08090134332
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 (1+p2T+p4T2)2 ( 1 + p^{2} T + p^{4} T^{2} )^{2}
3C2C_2 (1+p2T+p4T2)2 ( 1 + p^{2} T + p^{4} T^{2} )^{2}
7 1 1
good5D4×C2D_4\times C_2 1+53T1743T289994T3+2176954T489994p5T51743p10T6+53p15T7+p20T8 1 + 53 T - 1743 T^{2} - 89994 T^{3} + 2176954 T^{4} - 89994 p^{5} T^{5} - 1743 p^{10} T^{6} + 53 p^{15} T^{7} + p^{20} T^{8}
11D4×C2D_4\times C_2 1+191T234735T2883566pT3+345003244p2T4883566p6T5234735p10T6+191p15T7+p20T8 1 + 191 T - 234735 T^{2} - 883566 p T^{3} + 345003244 p^{2} T^{4} - 883566 p^{6} T^{5} - 234735 p^{10} T^{6} + 191 p^{15} T^{7} + p^{20} T^{8}
13D4D_{4} (1379T+488066T2379p5T3+p10T4)2 ( 1 - 379 T + 488066 T^{2} - 379 p^{5} T^{3} + p^{10} T^{4} )^{2}
17D4×C2D_4\times C_2 1+20pT1792914T218624000pT3+1462296318547T418624000p6T51792914p10T6+20p16T7+p20T8 1 + 20 p T - 1792914 T^{2} - 18624000 p T^{3} + 1462296318547 T^{4} - 18624000 p^{6} T^{5} - 1792914 p^{10} T^{6} + 20 p^{16} T^{7} + p^{20} T^{8}
19D4×C2D_4\times C_2 1+1769T+1045603T25074270360T39537626497976T45074270360p5T5+1045603p10T6+1769p15T7+p20T8 1 + 1769 T + 1045603 T^{2} - 5074270360 T^{3} - 9537626497976 T^{4} - 5074270360 p^{5} T^{5} + 1045603 p^{10} T^{6} + 1769 p^{15} T^{7} + p^{20} T^{8}
23D4×C2D_4\times C_2 1+3236T4980510T2+8347326720T3+129944731805059T4+8347326720p5T54980510p10T6+3236p15T7+p20T8 1 + 3236 T - 4980510 T^{2} + 8347326720 T^{3} + 129944731805059 T^{4} + 8347326720 p^{5} T^{5} - 4980510 p^{10} T^{6} + 3236 p^{15} T^{7} + p^{20} T^{8}
29D4D_{4} (14459T+37061638T24459p5T3+p10T4)2 ( 1 - 4459 T + 37061638 T^{2} - 4459 p^{5} T^{3} + p^{10} T^{4} )^{2}
31D4×C2D_4\times C_2 11994T+10280849T2+126744851310T3893708155041268T4+126744851310p5T5+10280849p10T61994p15T7+p20T8 1 - 1994 T + 10280849 T^{2} + 126744851310 T^{3} - 893708155041268 T^{4} + 126744851310 p^{5} T^{5} + 10280849 p^{10} T^{6} - 1994 p^{15} T^{7} + p^{20} T^{8}
37D4×C2D_4\times C_2 1+20587T+185423563T2+2052793425004T3+22636782601114654T4+2052793425004p5T5+185423563p10T6+20587p15T7+p20T8 1 + 20587 T + 185423563 T^{2} + 2052793425004 T^{3} + 22636782601114654 T^{4} + 2052793425004 p^{5} T^{5} + 185423563 p^{10} T^{6} + 20587 p^{15} T^{7} + p^{20} T^{8}
41D4D_{4} (1+8814T+240678562T2+8814p5T3+p10T4)2 ( 1 + 8814 T + 240678562 T^{2} + 8814 p^{5} T^{3} + p^{10} T^{4} )^{2}
43D4D_{4} (115853T+338678796T215853p5T3+p10T4)2 ( 1 - 15853 T + 338678796 T^{2} - 15853 p^{5} T^{3} + p^{10} T^{4} )^{2}
47D4×C2D_4\times C_2 133912T+462240278T27769017144224T3+156694760249296899T47769017144224p5T5+462240278p10T633912p15T7+p20T8 1 - 33912 T + 462240278 T^{2} - 7769017144224 T^{3} + 156694760249296899 T^{4} - 7769017144224 p^{5} T^{5} + 462240278 p^{10} T^{6} - 33912 p^{15} T^{7} + p^{20} T^{8}
53D4×C2D_4\times C_2 1+49239T+1103481011T2+23861570178636T3+556242294983257398T4+23861570178636p5T5+1103481011p10T6+49239p15T7+p20T8 1 + 49239 T + 1103481011 T^{2} + 23861570178636 T^{3} + 556242294983257398 T^{4} + 23861570178636 p^{5} T^{5} + 1103481011 p^{10} T^{6} + 49239 p^{15} T^{7} + p^{20} T^{8}
59D4×C2D_4\times C_2 1+56735T+988331391T2+45426593189460T3+2162900736067650880T4+45426593189460p5T5+988331391p10T6+56735p15T7+p20T8 1 + 56735 T + 988331391 T^{2} + 45426593189460 T^{3} + 2162900736067650880 T^{4} + 45426593189460 p^{5} T^{5} + 988331391 p^{10} T^{6} + 56735 p^{15} T^{7} + p^{20} T^{8}
61D4×C2D_4\times C_2 167508T+1731262802T276748134547280T3+3424209143194800779T476748134547280p5T5+1731262802p10T667508p15T7+p20T8 1 - 67508 T + 1731262802 T^{2} - 76748134547280 T^{3} + 3424209143194800779 T^{4} - 76748134547280 p^{5} T^{5} + 1731262802 p^{10} T^{6} - 67508 p^{15} T^{7} + p^{20} T^{8}
67D4×C2D_4\times C_2 1+75723T+1872823325T2+87906769364370T3+5344056371181586764T4+87906769364370p5T5+1872823325p10T6+75723p15T7+p20T8 1 + 75723 T + 1872823325 T^{2} + 87906769364370 T^{3} + 5344056371181586764 T^{4} + 87906769364370 p^{5} T^{5} + 1872823325 p^{10} T^{6} + 75723 p^{15} T^{7} + p^{20} T^{8}
71D4D_{4} (1+8992T681216182T2+8992p5T3+p10T4)2 ( 1 + 8992 T - 681216182 T^{2} + 8992 p^{5} T^{3} + p^{10} T^{4} )^{2}
73D4×C2D_4\times C_2 1+3201T2300766979T25874250509006T3+1021915489991879958T45874250509006p5T52300766979p10T6+3201p15T7+p20T8 1 + 3201 T - 2300766979 T^{2} - 5874250509006 T^{3} + 1021915489991879958 T^{4} - 5874250509006 p^{5} T^{5} - 2300766979 p^{10} T^{6} + 3201 p^{15} T^{7} + p^{20} T^{8}
79D4×C2D_4\times C_2 1+26612T3289450441T257387814991556T3+4333754463066999968T457387814991556p5T53289450441p10T6+26612p15T7+p20T8 1 + 26612 T - 3289450441 T^{2} - 57387814991556 T^{3} + 4333754463066999968 T^{4} - 57387814991556 p^{5} T^{5} - 3289450441 p^{10} T^{6} + 26612 p^{15} T^{7} + p^{20} T^{8}
83D4D_{4} (1949T+367057696T2949p5T3+p10T4)2 ( 1 - 949 T + 367057696 T^{2} - 949 p^{5} T^{3} + p^{10} T^{4} )^{2}
89D4×C2D_4\times C_2 1176562T+12318337010T21357352851108032T3+ 1 - 176562 T + 12318337010 T^{2} - 1357352851108032 T^{3} + 15 ⁣ ⁣9915\!\cdots\!99T41357352851108032p5T5+12318337010p10T6176562p15T7+p20T8 T^{4} - 1357352851108032 p^{5} T^{5} + 12318337010 p^{10} T^{6} - 176562 p^{15} T^{7} + p^{20} T^{8}
97D4D_{4} (1129423T+11942811256T2129423p5T3+p10T4)2 ( 1 - 129423 T + 11942811256 T^{2} - 129423 p^{5} T^{3} + p^{10} T^{4} )^{2}
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   L(s)=p j=18(1αj,pps)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

−7.60265475787005885523531859905, −7.50662608048168609205794364016, −7.31594869983354040218129889810, −6.96270304934182491780245541264, −6.75756850157364672983408263036, −6.16221680042365728170294524695, −6.11807911051165345421185958271, −6.02060327468357356532514031778, −5.91126277404953145765458783994, −5.04821341870029481448958778208, −4.89996542908997233300932179086, −4.80177503316309585991159397724, −4.56845285496617647628659917294, −4.17146902968841297812886795462, −3.73973190031628130907894059642, −3.42034549365550281877376625769, −3.38478337584057942038021712558, −2.57326640511419660239006376063, −2.49557265211285296202468567236, −1.88723688085889009487778301271, −1.60485374640931317454032050579, −1.08414002896829867392297372945, −0.823118429253004335813725830263, −0.50703506543731530270446824941, −0.092742387068855774462298142356, 0.092742387068855774462298142356, 0.50703506543731530270446824941, 0.823118429253004335813725830263, 1.08414002896829867392297372945, 1.60485374640931317454032050579, 1.88723688085889009487778301271, 2.49557265211285296202468567236, 2.57326640511419660239006376063, 3.38478337584057942038021712558, 3.42034549365550281877376625769, 3.73973190031628130907894059642, 4.17146902968841297812886795462, 4.56845285496617647628659917294, 4.80177503316309585991159397724, 4.89996542908997233300932179086, 5.04821341870029481448958778208, 5.91126277404953145765458783994, 6.02060327468357356532514031778, 6.11807911051165345421185958271, 6.16221680042365728170294524695, 6.75756850157364672983408263036, 6.96270304934182491780245541264, 7.31594869983354040218129889810, 7.50662608048168609205794364016, 7.60265475787005885523531859905

Graph of the ZZ-function along the critical line