↑

Properties

Label 8-650e4-1.1-c5e4-0-2
Degree 88
Conductor 178506250000178506250000
Sign 11
Analytic cond. 1.18112×1081.18112\times 10^{8}
Root an. cond. 10.210210.2102
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 32·4-s + 592·9-s − 72·11-s + 768·16-s + 2.18e3·19-s + 9.74e3·29-s − 4.52e3·31-s − 1.89e4·36-s − 1.84e4·41-s + 2.30e3·44-s + 2.42e4·49-s − 1.80e5·59-s − 1.38e5·61-s − 1.63e4·64-s − 5.73e4·71-s − 6.98e4·76-s − 1.68e5·79-s + 1.77e5·81-s − 1.85e5·89-s − 4.26e4·99-s + 7.78e3·101-s − 4.82e3·109-s − 3.11e5·116-s − 1.15e5·121-s + 1.44e5·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 2.43·9-s − 0.179·11-s + 3/4·16-s + 1.38·19-s + 2.15·29-s − 0.844·31-s − 2.43·36-s − 1.71·41-s + 0.179·44-s + 1.44·49-s − 6.76·59-s − 4.74·61-s − 1/2·64-s − 1.34·71-s − 1.38·76-s − 3.03·79-s + 2.99·81-s − 2.47·89-s − 0.437·99-s + 0.0759·101-s − 0.0388·109-s − 2.15·116-s − 0.716·121-s + 0.844·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

Functional equation

Λ(s)=((24⋅58⋅134)s/2 ΓC(s)4 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((24⋅58⋅134)s/2 ΓC(s+5/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 24⋅58⋅1342^{4} \cdot 5^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 1.18112×1081.18112\times 10^{8}
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 24⋅58⋅134, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) ≈\approx 0.93574291050.9357429105
L(12)L(\frac12) ≈\approx 0.93574291050.9357429105
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)
bad2C2C_2 (1+p4T2)2 ( 1 + p^{4} T^{2} )^{2}
5 1 1
13C2C_2 (1+p4T2)2 ( 1 + p^{4} T^{2} )^{2}
good3D4×C2D_4\times C_2 1−592T2+173458T4−592p10T6+p20T8 1 - 592 T^{2} + 173458 T^{4} - 592 p^{10} T^{6} + p^{20} T^{8}
7D4×C2D_4\times C_2 1−3468pT2+7278758p2T4−3468p11T6+p20T8 1 - 3468 p T^{2} + 7278758 p^{2} T^{4} - 3468 p^{11} T^{6} + p^{20} T^{8}
11D4D_{4} (1+36T+59660T2+36p5T3+p10T4)2 ( 1 + 36 T + 59660 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} )^{2}
17D4×C2D_4\times C_2 1−86540T2−3784960459802T4−86540p10T6+p20T8 1 - 86540 T^{2} - 3784960459802 T^{4} - 86540 p^{10} T^{6} + p^{20} T^{8}
19D4D_{4} (1−1092T+5226780T2−1092p5T3+p10T4)2 ( 1 - 1092 T + 5226780 T^{2} - 1092 p^{5} T^{3} + p^{10} T^{4} )^{2}
23D4×C2D_4\times C_2 1−4296896T2+73115643117858T4−4296896p10T6+p20T8 1 - 4296896 T^{2} + 73115643117858 T^{4} - 4296896 p^{10} T^{6} + p^{20} T^{8}
29D4D_{4} (1−168pT+24870890T2−168p6T3+p10T4)2 ( 1 - 168 p T + 24870890 T^{2} - 168 p^{6} T^{3} + p^{10} T^{4} )^{2}
31D4D_{4} (1+2260T−4802604T2+2260p5T3+p10T4)2 ( 1 + 2260 T - 4802604 T^{2} + 2260 p^{5} T^{3} + p^{10} T^{4} )^{2}
37D4×C2D_4\times C_2 1−81075300T2+5174214699551798T4−81075300p10T6+p20T8 1 - 81075300 T^{2} + 5174214699551798 T^{4} - 81075300 p^{10} T^{6} + p^{20} T^{8}
41D4D_{4} (1+9220T+191811102T2+9220p5T3+p10T4)2 ( 1 + 9220 T + 191811102 T^{2} + 9220 p^{5} T^{3} + p^{10} T^{4} )^{2}
43D4×C2D_4\times C_2 1−307121776T2+66493657362749938T4−307121776p10T6+p20T8 1 - 307121776 T^{2} + 66493657362749938 T^{4} - 307121776 p^{10} T^{6} + p^{20} T^{8}
47D4×C2D_4\times C_2 1−818687220T2+271437169988573798T4−818687220p10T6+p20T8 1 - 818687220 T^{2} + 271437169988573798 T^{4} - 818687220 p^{10} T^{6} + p^{20} T^{8}
53D4×C2D_4\times C_2 1−1603427228T2+992023260615384310T4−1603427228p10T6+p20T8 1 - 1603427228 T^{2} + 992023260615384310 T^{4} - 1603427228 p^{10} T^{6} + p^{20} T^{8}
59D4D_{4} (1+90404T+3449372988T2+90404p5T3+p10T4)2 ( 1 + 90404 T + 3449372988 T^{2} + 90404 p^{5} T^{3} + p^{10} T^{4} )^{2}
61D4D_{4} (1+69000T+2784511178T2+69000p5T3+p10T4)2 ( 1 + 69000 T + 2784511178 T^{2} + 69000 p^{5} T^{3} + p^{10} T^{4} )^{2}
67D4×C2D_4\times C_2 1−3438652308T2+5735146176860862518T4−3438652308p10T6+p20T8 1 - 3438652308 T^{2} + 5735146176860862518 T^{4} - 3438652308 p^{10} T^{6} + p^{20} T^{8}
71D4D_{4} (1+28668T+3332675492T2+28668p5T3+p10T4)2 ( 1 + 28668 T + 3332675492 T^{2} + 28668 p^{5} T^{3} + p^{10} T^{4} )^{2}
73D4×C2D_4\times C_2 1−6138404788T2+17643912066464393670T4−6138404788p10T6+p20T8 1 - 6138404788 T^{2} + 17643912066464393670 T^{4} - 6138404788 p^{10} T^{6} + p^{20} T^{8}
79D4D_{4} (1+84064T+4137892006T2+84064p5T3+p10T4)2 ( 1 + 84064 T + 4137892006 T^{2} + 84064 p^{5} T^{3} + p^{10} T^{4} )^{2}
83D4×C2D_4\times C_2 1−4727788228T2+35719446237480418710T4−4727788228p10T6+p20T8 1 - 4727788228 T^{2} + 35719446237480418710 T^{4} - 4727788228 p^{10} T^{6} + p^{20} T^{8}
89D4D_{4} (1+92620T+13143968022T2+92620p5T3+p10T4)2 ( 1 + 92620 T + 13143968022 T^{2} + 92620 p^{5} T^{3} + p^{10} T^{4} )^{2}
97D4×C2D_4\times C_2 1−28945542972T2+ 1 - 28945542972 T^{2} + 34 ⁣⋯ ⁣1034\!\cdots\!10T4−28945542972p10T6+p20T8 T^{4} - 28945542972 p^{10} T^{6} + p^{20} T^{8}
show more
show less
   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

−6.98757781218594973133115870854, −6.47624647882894459242544956580, −6.32225895189913788447326419851, −6.13172444207821644923400486541, −5.77644640849140991196864230450, −5.69599434135675158464679069133, −5.27502655931792704818662700044, −4.91076561495714424923936568113, −4.68940879937535060625406298549, −4.66197663612106823366654314072, −4.31966823979743706088062782619, −4.30542046913951442647374910370, −4.02481829338212262406214068930, −3.32659167346692373838370124474, −3.27596891446237640375953424171, −3.17746977429690549705256963460, −2.85084614691639572370735438477, −2.47354201959764268132148851212, −1.93858970345071507794288773128, −1.45490348744143616148447274799, −1.33035435032325176047832885419, −1.28978000147736598621030073539, −1.22022309103391759501140567243, −0.27551089067402655469312379277, −0.17792521344228198541589631241, 0.17792521344228198541589631241, 0.27551089067402655469312379277, 1.22022309103391759501140567243, 1.28978000147736598621030073539, 1.33035435032325176047832885419, 1.45490348744143616148447274799, 1.93858970345071507794288773128, 2.47354201959764268132148851212, 2.85084614691639572370735438477, 3.17746977429690549705256963460, 3.27596891446237640375953424171, 3.32659167346692373838370124474, 4.02481829338212262406214068930, 4.30542046913951442647374910370, 4.31966823979743706088062782619, 4.66197663612106823366654314072, 4.68940879937535060625406298549, 4.91076561495714424923936568113, 5.27502655931792704818662700044, 5.69599434135675158464679069133, 5.77644640849140991196864230450, 6.13172444207821644923400486541, 6.32225895189913788447326419851, 6.47624647882894459242544956580, 6.98757781218594973133115870854

Graph of the ZZ-function along the critical line