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Properties

Label 4-2254e2-1.1-c3e2-0-3
Degree 44
Conductor 50805165080516
Sign 11
Analytic cond. 17686.417686.4
Root an. cond. 11.532111.5321
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3·3-s + 12·4-s − 10·5-s − 12·6-s + 32·8-s − 29·9-s − 40·10-s − 12·11-s − 36·12-s + 51·13-s + 30·15-s + 80·16-s + 66·17-s − 116·18-s + 16·19-s − 120·20-s − 48·22-s + 46·23-s − 96·24-s − 102·25-s + 204·26-s + 120·27-s − 57·29-s + 120·30-s + 17·31-s + 192·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.894·5-s − 0.816·6-s + 1.41·8-s − 1.07·9-s − 1.26·10-s − 0.328·11-s − 0.866·12-s + 1.08·13-s + 0.516·15-s + 5/4·16-s + 0.941·17-s − 1.51·18-s + 0.193·19-s − 1.34·20-s − 0.465·22-s + 0.417·23-s − 0.816·24-s − 0.815·25-s + 1.53·26-s + 0.855·27-s − 0.364·29-s + 0.730·30-s + 0.0984·31-s + 1.06·32-s + ⋯

Functional equation

Λ(s)=(5080516s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 5080516 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(5080516s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5080516 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 50805165080516    =    22⋅74⋅2322^{2} \cdot 7^{4} \cdot 23^{2}
Sign: 11
Analytic conductor: 17686.417686.4
Root analytic conductor: 11.532111.5321
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 5080516, ( :3/2,3/2), 1)(4,\ 5080516,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad2C1C_1 (1−pT)2 ( 1 - p T )^{2}
7 1 1
23C1C_1 (1−pT)2 ( 1 - p T )^{2}
good3D4D_{4} 1+pT+38T2+p4T3+p6T4 1 + p T + 38 T^{2} + p^{4} T^{3} + p^{6} T^{4}
5D4D_{4} 1+2pT+202T2+2p4T3+p6T4 1 + 2 p T + 202 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4}
11C2C_2 (1+6T+p3T2)2 ( 1 + 6 T + p^{3} T^{2} )^{2}
13D4D_{4} 1−51T+2836T2−51p3T3+p6T4 1 - 51 T + 2836 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−66T+10258T2−66p3T3+p6T4 1 - 66 T + 10258 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−16T+6482T2−16p3T3+p6T4 1 - 16 T + 6482 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+57T+29716T2+57p3T3+p6T4 1 + 57 T + 29716 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−17T−27234T2−17p3T3+p6T4 1 - 17 T - 27234 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−206T+85562T2−206p3T3+p6T4 1 - 206 T + 85562 T^{2} - 206 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+373T+3836pT2+373p3T3+p6T4 1 + 373 T + 3836 p T^{2} + 373 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−314T+60950T2−314p3T3+p6T4 1 - 314 T + 60950 T^{2} - 314 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+859T+380710T2+859p3T3+p6T4 1 + 859 T + 380710 T^{2} + 859 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−50T+272026T2−50p3T3+p6T4 1 - 50 T + 272026 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+612T+438694T2+612p3T3+p6T4 1 + 612 T + 438694 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−1062T+674530T2−1062p3T3+p6T4 1 - 1062 T + 674530 T^{2} - 1062 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+844T+722378T2+844p3T3+p6T4 1 + 844 T + 722378 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−399T+747574T2−399p3T3+p6T4 1 - 399 T + 747574 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−1277T+1185552T2−1277p3T3+p6T4 1 - 1277 T + 1185552 T^{2} - 1277 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−122T+977462T2−122p3T3+p6T4 1 - 122 T + 977462 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1802T+1946542T2+1802p3T3+p6T4 1 + 1802 T + 1946542 T^{2} + 1802 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+2046T+2450554T2+2046p3T3+p6T4 1 + 2046 T + 2450554 T^{2} + 2046 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−910T+662234T2−910p3T3+p6T4 1 - 910 T + 662234 T^{2} - 910 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

−8.162889009552602294181453330613, −8.090536348939550719534202170578, −7.76188840686237989086919319308, −7.23751385200936398460061222504, −6.59626593570861254644306540182, −6.57579829511820589130392500583, −5.97672561447676353025157537566, −5.54919293971300598348733980328, −5.44465835973939616448873339390, −4.96798745089323424220283297200, −4.45672545316847206766154441900, −3.98215591874131931031467672338, −3.48453480355801118441674635196, −3.41980830657220066875069720155, −2.69906245147283147191011501672, −2.40673171734908573687266624004, −1.36458130653509153350932514836, −1.21576741934918205902719968133, 0, 0, 1.21576741934918205902719968133, 1.36458130653509153350932514836, 2.40673171734908573687266624004, 2.69906245147283147191011501672, 3.41980830657220066875069720155, 3.48453480355801118441674635196, 3.98215591874131931031467672338, 4.45672545316847206766154441900, 4.96798745089323424220283297200, 5.44465835973939616448873339390, 5.54919293971300598348733980328, 5.97672561447676353025157537566, 6.57579829511820589130392500583, 6.59626593570861254644306540182, 7.23751385200936398460061222504, 7.76188840686237989086919319308, 8.090536348939550719534202170578, 8.162889009552602294181453330613

Graph of the ZZ-function along the critical line