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Properties

Label 8-2400e4-1.1-c2e4-0-3
Degree 88
Conductor 3.318×10133.318\times 10^{13}
Sign 11
Analytic cond. 1.82887×1071.82887\times 10^{7}
Root an. cond. 8.086738.08673
Motivic weight 22
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  + 16·7-s + 6·9-s − 32·23-s + 40·29-s − 88·41-s − 160·43-s + 224·47-s + 60·49-s − 56·61-s + 96·63-s − 128·67-s + 27·81-s − 224·83-s + 312·89-s + 664·101-s + 16·103-s − 192·107-s + 24·109-s + 260·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 512·161-s + ⋯
L(s)  = 1  + 16/7·7-s + 2/3·9-s − 1.39·23-s + 1.37·29-s − 2.14·41-s − 3.72·43-s + 4.76·47-s + 1.22·49-s − 0.918·61-s + 1.52·63-s − 1.91·67-s + 1/3·81-s − 2.69·83-s + 3.50·89-s + 6.57·101-s + 0.155·103-s − 1.79·107-s + 0.220·109-s + 2.14·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 3.18·161-s + ⋯

Functional equation

Λ(s)=((220⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(3−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(3-s)\end{aligned}
Λ(s)=((220⋅34⋅58)s/2 ΓC(s+1)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)^{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: 1.82887×1071.82887\times 10^{7}
Root analytic conductor: 8.086738.08673
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅34⋅58, ( :1,1,1,1), 1)(8,\ 2^{20} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 7.8852001747.885200174
L(12)L(\frac12) ≈\approx 7.8852001747.885200174
L(2)L(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
3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5 1 1
good7D4D_{4} (1−8T+66T2−8p2T3+p4T4)2 ( 1 - 8 T + 66 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2}
11D4×C2D_4\times C_2 1−260T2+33894T4−260p4T6+p8T8 1 - 260 T^{2} + 33894 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8}
13D4×C2D_4\times C_2 1−92T2−17562T4−92p4T6+p8T8 1 - 92 T^{2} - 17562 T^{4} - 92 p^{4} T^{6} + p^{8} T^{8}
17D4×C2D_4\times C_2 1−700T2+261894T4−700p4T6+p8T8 1 - 700 T^{2} + 261894 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8}
19D4×C2D_4\times C_2 1−196T2+159654T4−196p4T6+p8T8 1 - 196 T^{2} + 159654 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8}
23C2C_2 (1+8T+p2T2)4 ( 1 + 8 T + p^{2} T^{2} )^{4}
29D4D_{4} (1−20T+1734T2−20p2T3+p4T4)2 ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
31D4×C2D_4\times C_2 1−1412T2+2268678T4−1412p4T6+p8T8 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8}
37D4×C2D_4\times C_2 1−3292T2+3990p2T4−3292p4T6+p8T8 1 - 3292 T^{2} + 3990 p^{2} T^{4} - 3292 p^{4} T^{6} + p^{8} T^{8}
41D4D_{4} (1+44T+3654T2+44p2T3+p4T4)2 ( 1 + 44 T + 3654 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4D_{4} (1+80T+4098T2+80p2T3+p4T4)2 ( 1 + 80 T + 4098 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2}
47D4D_{4} (1−112T+7362T2−112p2T3+p4T4)2 ( 1 - 112 T + 7362 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2}
53D4×C2D_4\times C_2 1−11068T2+46399206T4−11068p4T6+p8T8 1 - 11068 T^{2} + 46399206 T^{4} - 11068 p^{4} T^{6} + p^{8} T^{8}
59D4×C2D_4\times C_2 1−260T2+462054T4−260p4T6+p8T8 1 - 260 T^{2} + 462054 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8}
61C2C_2 (1+14T+p2T2)4 ( 1 + 14 T + p^{2} T^{2} )^{4}
67D4D_{4} (1+64T+7650T2+64p2T3+p4T4)2 ( 1 + 64 T + 7650 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2}
71D4×C2D_4\times C_2 1−3140T2+9051462T4−3140p4T6+p8T8 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8}
73D4×C2D_4\times C_2 1−13372T2+90439878T4−13372p4T6+p8T8 1 - 13372 T^{2} + 90439878 T^{4} - 13372 p^{4} T^{6} + p^{8} T^{8}
79D4×C2D_4\times C_2 1−12548T2+107282310T4−12548p4T6+p8T8 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8}
83D4D_{4} (1+112T+16866T2+112p2T3+p4T4)2 ( 1 + 112 T + 16866 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{2}
89D4D_{4} (1−156T+21158T2−156p2T3+p4T4)2 ( 1 - 156 T + 21158 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4×C2D_4\times C_2 1−16124T2+135775494T4−16124p4T6+p8T8 1 - 16124 T^{2} + 135775494 T^{4} - 16124 p^{4} T^{6} + p^{8} 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

−6.12451941926999495325739475960, −6.03208844416074999839953101512, −5.82852508366691041337577411083, −5.48498136375244718295027976706, −5.24840496466630231820128582590, −4.99025064222820037379099235186, −4.82537805946898397638851746399, −4.77212974632791444277020069244, −4.59390084572989829555375052813, −4.46380409453654099556027639396, −4.00967924110407174899875285826, −3.85578627108723818195243741431, −3.62315504833312831281628753005, −3.40208833957559305566180680404, −3.19306688227608737086726324480, −2.82183303690798302984447902779, −2.50915735469295351129733853014, −2.25445218293342034125111943773, −1.91518501642645343230119364302, −1.87529242680191164721104249201, −1.52680392142717275354575687927, −1.30207589979741729554202849243, −1.08084890569095120765143974585, −0.46887031296433922842496471490, −0.37735649007442704438870130506, 0.37735649007442704438870130506, 0.46887031296433922842496471490, 1.08084890569095120765143974585, 1.30207589979741729554202849243, 1.52680392142717275354575687927, 1.87529242680191164721104249201, 1.91518501642645343230119364302, 2.25445218293342034125111943773, 2.50915735469295351129733853014, 2.82183303690798302984447902779, 3.19306688227608737086726324480, 3.40208833957559305566180680404, 3.62315504833312831281628753005, 3.85578627108723818195243741431, 4.00967924110407174899875285826, 4.46380409453654099556027639396, 4.59390084572989829555375052813, 4.77212974632791444277020069244, 4.82537805946898397638851746399, 4.99025064222820037379099235186, 5.24840496466630231820128582590, 5.48498136375244718295027976706, 5.82852508366691041337577411083, 6.03208844416074999839953101512, 6.12451941926999495325739475960

Graph of the ZZ-function along the critical line