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Properties

Label 8-2400e4-1.1-c1e4-0-12
Degree 88
Conductor 3.318×10133.318\times 10^{13}
Sign 11
Analytic cond. 134881.134881.
Root an. cond. 4.377684.37768
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 8·9-s + 8·13-s − 24·23-s − 12·27-s + 24·37-s − 32·39-s + 8·47-s + 16·49-s − 16·59-s + 96·69-s + 16·71-s − 40·73-s + 23·81-s + 8·83-s − 8·97-s + 56·107-s + 48·109-s − 96·111-s + 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s − 32·141-s − 64·147-s + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s + 2.21·13-s − 5.00·23-s − 2.30·27-s + 3.94·37-s − 5.12·39-s + 1.16·47-s + 16/7·49-s − 2.08·59-s + 11.5·69-s + 1.89·71-s − 4.68·73-s + 23/9·81-s + 0.878·83-s − 0.812·97-s + 5.41·107-s + 4.59·109-s − 9.11·111-s + 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.69·141-s − 5.27·147-s + ⋯

Functional equation

Λ(s)=((220⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅34⋅58)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅34⋅582^{20} \cdot 3^{4} \cdot 5^{8}
Sign: 11
Analytic conductor: 134881.134881.
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅34⋅58, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.84738100950.8473810095
L(12)L(\frac12) ≈\approx 0.84738100950.8473810095
L(32)L(\frac{3}{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)
bad2 1 1
3C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
5 1 1
good7D4×C2D_4\times C_2 1−16T2+130T4−16p2T6+p4T8 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
11C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
17C4×C2C_4\times C_2 1+4T2+70T4+4p2T6+p4T8 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
23D4D_{4} (1+12T+80T2+12pT3+p2T4)2 ( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
37D4D_{4} (1−12T+78T2−12pT3+p2T4)2 ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
41C22C_2^2 (1−78T2+p2T4)2 ( 1 - 78 T^{2} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+32T2+2386T4+32p2T6+p4T8 1 + 32 T^{2} + 2386 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1−4T+80T2−4pT3+p2T4)2 ( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−140T2+10006T4−140p2T6+p4T8 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1+8T+102T2+8pT3+p2T4)2 ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+90T2+p2T4)2 ( 1 + 90 T^{2} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−96T2+4082T4−96p2T6+p4T8 1 - 96 T^{2} + 4082 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−8T+150T2−8pT3+p2T4)2 ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
73C4C_4 (1+20T+214T2+20pT3+p2T4)2 ( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−140T2+12774T4−140p2T6+p4T8 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
83D4D_{4} (1−4T+120T2−4pT3+p2T4)2 ( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1−68T2+8806T4−68p2T6+p4T8 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}
97C4C_4 (1+4T−90T2+4pT3+p2T4)2 ( 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
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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

−6.17713149860678643331383221510, −6.10845405453645328375740149373, −6.03307106001452051527081657612, −5.98754320303236535099207583758, −5.74352270801399392139480721057, −5.38245422755165215201302202957, −5.26374552899452608100256393839, −4.88336318225739096748002627665, −4.77805871966407091685237281692, −4.27482419271497037993503947796, −4.15407567677816896205566677103, −4.15051117965543369882137859568, −4.09639996082715371386178623933, −3.73802989254181979652992466693, −3.43392920585577862370235261903, −3.09404254666291495435056255886, −2.87026187545546930864971226942, −2.41219003025907057732557534086, −2.09894035753398415400069256035, −2.03423143976675607824762267787, −1.70051347992213991493081183780, −1.18943428194336203479094081616, −0.999532709386729663943355767692, −0.63659813908560042971152972401, −0.25946295004829184477291133831, 0.25946295004829184477291133831, 0.63659813908560042971152972401, 0.999532709386729663943355767692, 1.18943428194336203479094081616, 1.70051347992213991493081183780, 2.03423143976675607824762267787, 2.09894035753398415400069256035, 2.41219003025907057732557534086, 2.87026187545546930864971226942, 3.09404254666291495435056255886, 3.43392920585577862370235261903, 3.73802989254181979652992466693, 4.09639996082715371386178623933, 4.15051117965543369882137859568, 4.15407567677816896205566677103, 4.27482419271497037993503947796, 4.77805871966407091685237281692, 4.88336318225739096748002627665, 5.26374552899452608100256393839, 5.38245422755165215201302202957, 5.74352270801399392139480721057, 5.98754320303236535099207583758, 6.03307106001452051527081657612, 6.10845405453645328375740149373, 6.17713149860678643331383221510

Graph of the ZZ-function along the critical line