↑

Properties

Label 4-38e2-1.1-c3e2-0-1
Degree 44
Conductor 14441444
Sign 11
Analytic cond. 5.026885.02688
Root an. cond. 1.497351.49735
Motivic weight 33
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  − 4·2-s + 3-s + 12·4-s + 10·5-s − 4·6-s + 57·7-s − 32·8-s − 9·9-s − 40·10-s + 10·11-s + 12·12-s + 13·13-s − 228·14-s + 10·15-s + 80·16-s − 51·17-s + 36·18-s − 38·19-s + 120·20-s + 57·21-s − 40·22-s − 155·23-s − 32·24-s + 2·25-s − 52·26-s + 8·27-s + 684·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.192·3-s + 3/2·4-s + 0.894·5-s − 0.272·6-s + 3.07·7-s − 1.41·8-s − 1/3·9-s − 1.26·10-s + 0.274·11-s + 0.288·12-s + 0.277·13-s − 4.35·14-s + 0.172·15-s + 5/4·16-s − 0.727·17-s + 0.471·18-s − 0.458·19-s + 1.34·20-s + 0.592·21-s − 0.387·22-s − 1.40·23-s − 0.272·24-s + 0.0159·25-s − 0.392·26-s + 0.0570·27-s + 4.61·28-s + ⋯

Functional equation

Λ(s)=(1444s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1444s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14441444    =    22⋅1922^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 5.026885.02688
Root analytic conductor: 1.497351.49735
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1444, ( :3/2,3/2), 1)(4,\ 1444,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.2619197951.261919795
L(12)L(\frac12) ≈\approx 1.2619197951.261919795
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}
19C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3D4D_{4} 1−T+10T2−p3T3+p6T4 1 - T + 10 T^{2} - p^{3} T^{3} + p^{6} T^{4}
5D4D_{4} 1−2pT+98T2−2p4T3+p6T4 1 - 2 p T + 98 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4}
7D4D_{4} 1−57T+1454T2−57p3T3+p6T4 1 - 57 T + 1454 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−10T+2510T2−10p3T3+p6T4 1 - 10 T + 2510 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−pT+2268T2−p4T3+p6T4 1 - p T + 2268 T^{2} - p^{4} T^{3} + p^{6} T^{4}
17D4D_{4} 1+3pT+520T2+3p4T3+p6T4 1 + 3 p T + 520 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4}
23D4D_{4} 1+155T+994pT2+155p3T3+p6T4 1 + 155 T + 994 p T^{2} + 155 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+79T+13124T2+79p3T3+p6T4 1 + 79 T + 13124 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+16T+48318T2+16p3T3+p6T4 1 + 16 T + 48318 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−380T+126078T2−380p3T3+p6T4 1 - 380 T + 126078 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+790T+292274T2+790p3T3+p6T4 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−296T+78966T2−296p3T3+p6T4 1 - 296 T + 78966 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+200T+146846T2+200p3T3+p6T4 1 + 200 T + 146846 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−397T+333572T2−397p3T3+p6T4 1 - 397 T + 333572 T^{2} - 397 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−201T+197794T2−201p3T3+p6T4 1 - 201 T + 197794 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+680T+483894T2+680p3T3+p6T4 1 + 680 T + 483894 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+939T+740138T2+939p3T3+p6T4 1 + 939 T + 740138 T^{2} + 939 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−406T+735614T2−406p3T3+p6T4 1 - 406 T + 735614 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−123T+781772T2−123p3T3+p6T4 1 - 123 T + 781772 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−106T+840030T2−106p3T3+p6T4 1 - 106 T + 840030 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−2226T+2380750T2−2226p3T3+p6T4 1 - 2226 T + 2380750 T^{2} - 2226 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+870T+1594738T2+870p3T3+p6T4 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1864T+2438382T2+1864p3T3+p6T4 1 + 1864 T + 2438382 T^{2} + 1864 p^{3} T^{3} + p^{6} T^{4}
show more
show less
   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

−16.42362552439556254127211450200, −15.44249293269138385164643740783, −14.83341150220674979248969608990, −14.65462673527854666915208766640, −13.84081462250573773073310298309, −13.42805270694816707438894980498, −11.93686823526188039545269815241, −11.81518792602670282986256710435, −10.89036109132546340838310197267, −10.80785125592809109236580824559, −9.841794763631763200053699624608, −9.072963783034075895055675876009, −8.266695078387097207180441493137, −8.232195932100640022413740604025, −7.35742288151623467179702724045, −6.24615267287495912324842548824, −5.39204263415166716592434760149, −4.37425361656522483900048009165, −2.14164103742936405307762083682, −1.58422284959852460289428135042, 1.58422284959852460289428135042, 2.14164103742936405307762083682, 4.37425361656522483900048009165, 5.39204263415166716592434760149, 6.24615267287495912324842548824, 7.35742288151623467179702724045, 8.232195932100640022413740604025, 8.266695078387097207180441493137, 9.072963783034075895055675876009, 9.841794763631763200053699624608, 10.80785125592809109236580824559, 10.89036109132546340838310197267, 11.81518792602670282986256710435, 11.93686823526188039545269815241, 13.42805270694816707438894980498, 13.84081462250573773073310298309, 14.65462673527854666915208766640, 14.83341150220674979248969608990, 15.44249293269138385164643740783, 16.42362552439556254127211450200

Graph of the ZZ-function along the critical line