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Properties

Label 4-816e2-1.1-c3e2-0-2
Degree 44
Conductor 665856665856
Sign 11
Analytic cond. 2317.992317.99
Root an. cond. 6.938706.93870
Motivic weight 33
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  + 6·3-s + 3·5-s + 2·7-s + 27·9-s + 15·11-s − 23·13-s + 18·15-s − 34·17-s + 137·19-s + 12·21-s + 201·23-s − 193·25-s + 108·27-s − 66·29-s + 464·31-s + 90·33-s + 6·35-s − 140·37-s − 138·39-s − 45·41-s + 407·43-s + 81·45-s + 114·47-s − 482·49-s − 204·51-s + 792·53-s + 45·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.268·5-s + 0.107·7-s + 9-s + 0.411·11-s − 0.490·13-s + 0.309·15-s − 0.485·17-s + 1.65·19-s + 0.124·21-s + 1.82·23-s − 1.54·25-s + 0.769·27-s − 0.422·29-s + 2.68·31-s + 0.474·33-s + 0.0289·35-s − 0.622·37-s − 0.566·39-s − 0.171·41-s + 1.44·43-s + 0.268·45-s + 0.353·47-s − 1.40·49-s − 0.560·51-s + 2.05·53-s + 0.110·55-s + ⋯

Functional equation

Λ(s)=(665856s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(665856s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 665856665856    =    28⋅32⋅1722^{8} \cdot 3^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 2317.992317.99
Root analytic conductor: 6.938706.93870
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 665856, ( :3/2,3/2), 1)(4,\ 665856,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 6.6075089176.607508917
L(12)L(\frac12) ≈\approx 6.6075089176.607508917
L(52)L(\frac{5}{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
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
17C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1−3T+202T2−3p3T3+p6T4 1 - 3 T + 202 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−2T+486T2−2p3T3+p6T4 1 - 2 T + 486 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−15T+1462T2−15p3T3+p6T4 1 - 15 T + 1462 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+23T+2064T2+23p3T3+p6T4 1 + 23 T + 2064 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−137T+17154T2−137p3T3+p6T4 1 - 137 T + 17154 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−201T+34384T2−201p3T3+p6T4 1 - 201 T + 34384 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+66T+25546T2+66p3T3+p6T4 1 + 66 T + 25546 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−464T+106170T2−464p3T3+p6T4 1 - 464 T + 106170 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+140T−29670T2+140p3T3+p6T4 1 + 140 T - 29670 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+45T+36592T2+45p3T3+p6T4 1 + 45 T + 36592 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−407T+178266T2−407p3T3+p6T4 1 - 407 T + 178266 T^{2} - 407 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−114T+85270T2−114p3T3+p6T4 1 - 114 T + 85270 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−792T+357286T2−792p3T3+p6T4 1 - 792 T + 357286 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+570T+467662T2+570p3T3+p6T4 1 + 570 T + 467662 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−16T+248202T2−16p3T3+p6T4 1 - 16 T + 248202 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+208T+454758T2+208p3T3+p6T4 1 + 208 T + 454758 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+90T+716038T2+90p3T3+p6T4 1 + 90 T + 716038 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4}
73C2C_2 (1−2pT+p3T2)2 ( 1 - 2 p T + p^{3} T^{2} )^{2}
79D4D_{4} 1−764T+1103058T2−764p3T3+p6T4 1 - 764 T + 1103058 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−138T+902110T2−138p3T3+p6T4 1 - 138 T + 902110 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−654T+768946T2−654p3T3+p6T4 1 - 654 T + 768946 T^{2} - 654 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+146T+1082754T2+146p3T3+p6T4 1 + 146 T + 1082754 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.850418641384548696527887464491, −9.597714813829746035281433612415, −9.074486400134581859404249463038, −9.074216963785299372071046174557, −8.295058783412092579409037897190, −8.006734036988850856298238972935, −7.44148831419494020915863749289, −7.30313087754014174763733251713, −6.55763512103065883410628321829, −6.37822350659271709435643637579, −5.45281473529071606060254848236, −5.27631944498929056491434767062, −4.39591696797228239335789770896, −4.34344323047402360931541384687, −3.36874161883012455346597017295, −3.14936149689019291169557141968, −2.50595583320225255017454040756, −1.99944146754355604165535378692, −1.20745149444770205506083954523, −0.69356551879258903859245915809, 0.69356551879258903859245915809, 1.20745149444770205506083954523, 1.99944146754355604165535378692, 2.50595583320225255017454040756, 3.14936149689019291169557141968, 3.36874161883012455346597017295, 4.34344323047402360931541384687, 4.39591696797228239335789770896, 5.27631944498929056491434767062, 5.45281473529071606060254848236, 6.37822350659271709435643637579, 6.55763512103065883410628321829, 7.30313087754014174763733251713, 7.44148831419494020915863749289, 8.006734036988850856298238972935, 8.295058783412092579409037897190, 9.074216963785299372071046174557, 9.074486400134581859404249463038, 9.597714813829746035281433612415, 9.850418641384548696527887464491

Graph of the ZZ-function along the critical line