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Properties

Label 8-490e4-1.1-c5e4-0-3
Degree 88
Conductor 5764801000057648010000
Sign 11
Analytic cond. 3.81440×1073.81440\times 10^{7}
Root an. cond. 8.864998.86499
Motivic weight 55
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  + 16·2-s − 40·3-s + 160·4-s + 100·5-s − 640·6-s + 1.28e3·8-s + 450·9-s + 1.60e3·10-s − 432·11-s − 6.40e3·12-s − 620·13-s − 4.00e3·15-s + 8.96e3·16-s − 2.26e3·17-s + 7.20e3·18-s − 1.48e3·19-s + 1.60e4·20-s − 6.91e3·22-s + 1.96e3·23-s − 5.12e4·24-s + 6.25e3·25-s − 9.92e3·26-s + 3.28e3·27-s − 564·29-s − 6.40e4·30-s − 1.02e4·31-s + 5.73e4·32-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.56·3-s + 5·4-s + 1.78·5-s − 7.25·6-s + 7.07·8-s + 1.85·9-s + 5.05·10-s − 1.07·11-s − 12.8·12-s − 1.01·13-s − 4.59·15-s + 35/4·16-s − 1.89·17-s + 5.23·18-s − 0.940·19-s + 8.94·20-s − 3.04·22-s + 0.775·23-s − 18.1·24-s + 2·25-s − 2.87·26-s + 0.865·27-s − 0.124·29-s − 12.9·30-s − 1.91·31-s + 9.89·32-s + ⋯

Functional equation

Λ(s)=((24⋅54⋅78)s/2 ΓC(s)4 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((24⋅54⋅78)s/2 ΓC(s+5/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{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: 24⋅54⋅782^{4} \cdot 5^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 3.81440×1073.81440\times 10^{7}
Root analytic conductor: 8.864998.86499
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 24⋅54⋅78, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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)
bad2C1C_1 (1−p2T)4 ( 1 - p^{2} T )^{4}
5C1C_1 (1−p2T)4 ( 1 - p^{2} T )^{4}
7 1 1
good3(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+40T+1150T2+8240pT3+48803p2T4+8240p6T5+1150p10T6+40p15T7+p20T8 1 + 40 T + 1150 T^{2} + 8240 p T^{3} + 48803 p^{2} T^{4} + 8240 p^{6} T^{5} + 1150 p^{10} T^{6} + 40 p^{15} T^{7} + p^{20} T^{8}
11(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+432T+424946T2+155685072T3+100205028739T4+155685072p5T5+424946p10T6+432p15T7+p20T8 1 + 432 T + 424946 T^{2} + 155685072 T^{3} + 100205028739 T^{4} + 155685072 p^{5} T^{5} + 424946 p^{10} T^{6} + 432 p^{15} T^{7} + p^{20} T^{8}
13(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+620T+1364814T2+630975960T3+736096010131T4+630975960p5T5+1364814p10T6+620p15T7+p20T8 1 + 620 T + 1364814 T^{2} + 630975960 T^{3} + 736096010131 T^{4} + 630975960 p^{5} T^{5} + 1364814 p^{10} T^{6} + 620 p^{15} T^{7} + p^{20} T^{8}
17(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+2260T+6689558T2+9325180360T3+14902549646075T4+9325180360p5T5+6689558p10T6+2260p15T7+p20T8 1 + 2260 T + 6689558 T^{2} + 9325180360 T^{3} + 14902549646075 T^{4} + 9325180360 p^{5} T^{5} + 6689558 p^{10} T^{6} + 2260 p^{15} T^{7} + p^{20} T^{8}
19(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+1480T+9396964T2+10144231720T3+34421947075670T4+10144231720p5T5+9396964p10T6+1480p15T7+p20T8 1 + 1480 T + 9396964 T^{2} + 10144231720 T^{3} + 34421947075670 T^{4} + 10144231720 p^{5} T^{5} + 9396964 p^{10} T^{6} + 1480 p^{15} T^{7} + p^{20} T^{8}
23(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1−1968T+9987488T2−32046096912T3+52040993557570T4−32046096912p5T5+9987488p10T6−1968p15T7+p20T8 1 - 1968 T + 9987488 T^{2} - 32046096912 T^{3} + 52040993557570 T^{4} - 32046096912 p^{5} T^{5} + 9987488 p^{10} T^{6} - 1968 p^{15} T^{7} + p^{20} T^{8}
29(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+564T+69433474T2+9006757936T3+1994184496605979T4+9006757936p5T5+69433474p10T6+564p15T7+p20T8 1 + 564 T + 69433474 T^{2} + 9006757936 T^{3} + 1994184496605979 T^{4} + 9006757936 p^{5} T^{5} + 69433474 p^{10} T^{6} + 564 p^{15} T^{7} + p^{20} T^{8}
31(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+10240T+148124560T2+29604903040pT3+6856056611517602T4+29604903040p6T5+148124560p10T6+10240p15T7+p20T8 1 + 10240 T + 148124560 T^{2} + 29604903040 p T^{3} + 6856056611517602 T^{4} + 29604903040 p^{6} T^{5} + 148124560 p^{10} T^{6} + 10240 p^{15} T^{7} + p^{20} T^{8}
37(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+1648T+161896744T2+101725654768T3+13339828319781218T4+101725654768p5T5+161896744p10T6+1648p15T7+p20T8 1 + 1648 T + 161896744 T^{2} + 101725654768 T^{3} + 13339828319781218 T^{4} + 101725654768 p^{5} T^{5} + 161896744 p^{10} T^{6} + 1648 p^{15} T^{7} + p^{20} T^{8}
41(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+32320T+609845624T2+7548797241280T3+84857018876077202T4+7548797241280p5T5+609845624p10T6+32320p15T7+p20T8 1 + 32320 T + 609845624 T^{2} + 7548797241280 T^{3} + 84857018876077202 T^{4} + 7548797241280 p^{5} T^{5} + 609845624 p^{10} T^{6} + 32320 p^{15} T^{7} + p^{20} T^{8}
43(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1−19320T+647664324T2−8553439555800T3+147201745308735542T4−8553439555800p5T5+647664324p10T6−19320p15T7+p20T8 1 - 19320 T + 647664324 T^{2} - 8553439555800 T^{3} + 147201745308735542 T^{4} - 8553439555800 p^{5} T^{5} + 647664324 p^{10} T^{6} - 19320 p^{15} T^{7} + p^{20} T^{8}
47(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+15600T+589104950T2+3062232126720T3+126459226732351027T4+3062232126720p5T5+589104950p10T6+15600p15T7+p20T8 1 + 15600 T + 589104950 T^{2} + 3062232126720 T^{3} + 126459226732351027 T^{4} + 3062232126720 p^{5} T^{5} + 589104950 p^{10} T^{6} + 15600 p^{15} T^{7} + p^{20} T^{8}
53(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1−28928T+1635333216T2−31852579536384T3+1005064993135837554T4−31852579536384p5T5+1635333216p10T6−28928p15T7+p20T8 1 - 28928 T + 1635333216 T^{2} - 31852579536384 T^{3} + 1005064993135837554 T^{4} - 31852579536384 p^{5} T^{5} + 1635333216 p^{10} T^{6} - 28928 p^{15} T^{7} + p^{20} T^{8}
59(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+5800T−228430544T2+9932648143800T3+786538339437354002T4+9932648143800p5T5−228430544p10T6+5800p15T7+p20T8 1 + 5800 T - 228430544 T^{2} + 9932648143800 T^{3} + 786538339437354002 T^{4} + 9932648143800 p^{5} T^{5} - 228430544 p^{10} T^{6} + 5800 p^{15} T^{7} + p^{20} T^{8}
61(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+83720T+4627195636T2+172113870704120T3+5572286308486140470T4+172113870704120p5T5+4627195636p10T6+83720p15T7+p20T8 1 + 83720 T + 4627195636 T^{2} + 172113870704120 T^{3} + 5572286308486140470 T^{4} + 172113870704120 p^{5} T^{5} + 4627195636 p^{10} T^{6} + 83720 p^{15} T^{7} + p^{20} T^{8}
67(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+33000T+5259529240T2+125466399418200T3+10527764775076193698T4+125466399418200p5T5+5259529240p10T6+33000p15T7+p20T8 1 + 33000 T + 5259529240 T^{2} + 125466399418200 T^{3} + 10527764775076193698 T^{4} + 125466399418200 p^{5} T^{5} + 5259529240 p^{10} T^{6} + 33000 p^{15} T^{7} + p^{20} T^{8}
71(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1−4112T+162751908T2−37355181830544T3+877858340675496870T4−37355181830544p5T5+162751908p10T6−4112p15T7+p20T8 1 - 4112 T + 162751908 T^{2} - 37355181830544 T^{3} + 877858340675496870 T^{4} - 37355181830544 p^{5} T^{5} + 162751908 p^{10} T^{6} - 4112 p^{15} T^{7} + p^{20} T^{8}
73(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+84400T+8395914132T2+436767418008080T3+25012278932128479110T4+436767418008080p5T5+8395914132p10T6+84400p15T7+p20T8 1 + 84400 T + 8395914132 T^{2} + 436767418008080 T^{3} + 25012278932128479110 T^{4} + 436767418008080 p^{5} T^{5} + 8395914132 p^{10} T^{6} + 84400 p^{15} T^{7} + p^{20} T^{8}
79(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+44648T+3915859882T2+198304071878096T3+23368298036236933259T4+198304071878096p5T5+3915859882p10T6+44648p15T7+p20T8 1 + 44648 T + 3915859882 T^{2} + 198304071878096 T^{3} + 23368298036236933259 T^{4} + 198304071878096 p^{5} T^{5} + 3915859882 p^{10} T^{6} + 44648 p^{15} T^{7} + p^{20} T^{8}
83(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+129120T+20912281996T2+1580873942983520T3+ 1 + 129120 T + 20912281996 T^{2} + 1580873942983520 T^{3} + 13 ⁣⋯ ⁣0213\!\cdots\!02T4+1580873942983520p5T5+20912281996p10T6+129120p15T7+p20T8 T^{4} + 1580873942983520 p^{5} T^{5} + 20912281996 p^{10} T^{6} + 129120 p^{15} T^{7} + p^{20} T^{8}
89(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1+309800T+51469973748T2+5970269052356280T3+ 1 + 309800 T + 51469973748 T^{2} + 5970269052356280 T^{3} + 51 ⁣⋯ ⁣0251\!\cdots\!02T4+5970269052356280p5T5+51469973748p10T6+309800p15T7+p20T8 T^{4} + 5970269052356280 p^{5} T^{5} + 51469973748 p^{10} T^{6} + 309800 p^{15} T^{7} + p^{20} T^{8}
97(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2 1−5220T+27657295382T2+978769401240T3+ 1 - 5220 T + 27657295382 T^{2} + 978769401240 T^{3} + 32 ⁣⋯ ⁣7932\!\cdots\!79T4+978769401240p5T5+27657295382p10T6−5220p15T7+p20T8 T^{4} + 978769401240 p^{5} T^{5} + 27657295382 p^{10} T^{6} - 5220 p^{15} T^{7} + 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

−7.28413635820975987062670514036, −6.84157712582858095073167660077, −6.80543587330803708486577057578, −6.60067949523966819854660640013, −6.58491029432711012933765817567, −6.00794894804152440657068035185, −5.92647385097282771125121157206, −5.76521854140670405978798278322, −5.63813169990461271423590345064, −5.22052750822711979374558044446, −5.10117840994094939324189245453, −5.05137395462990635189800138691, −4.98446190867513019323673349323, −4.35522627729635474817724807778, −4.22847239940124644770689890196, −3.91800154982087090414175867425, −3.73275220912251184955501548713, −2.76152061352543979186966655918, −2.75085905274412054897301435172, −2.73275084334761361485536229703, −2.71102440614718416731050357358, −1.87764276777206850481614960163, −1.71985010280181465723176865392, −1.39689279777401256275828606329, −1.26596724798542388294792693606, 0, 0, 0, 0, 1.26596724798542388294792693606, 1.39689279777401256275828606329, 1.71985010280181465723176865392, 1.87764276777206850481614960163, 2.71102440614718416731050357358, 2.73275084334761361485536229703, 2.75085905274412054897301435172, 2.76152061352543979186966655918, 3.73275220912251184955501548713, 3.91800154982087090414175867425, 4.22847239940124644770689890196, 4.35522627729635474817724807778, 4.98446190867513019323673349323, 5.05137395462990635189800138691, 5.10117840994094939324189245453, 5.22052750822711979374558044446, 5.63813169990461271423590345064, 5.76521854140670405978798278322, 5.92647385097282771125121157206, 6.00794894804152440657068035185, 6.58491029432711012933765817567, 6.60067949523966819854660640013, 6.80543587330803708486577057578, 6.84157712582858095073167660077, 7.28413635820975987062670514036

Graph of the ZZ-function along the critical line