Properties

Label 4-950e2-1.1-c5e2-0-1
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 23214.923214.9
Root an. cond. 12.343612.3436
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 3·3-s + 48·4-s − 24·6-s − 114·7-s + 256·8-s − 119·9-s + 661·11-s − 144·12-s − 1.61e3·13-s − 912·14-s + 1.28e3·16-s − 64·17-s − 952·18-s + 722·19-s + 342·21-s + 5.28e3·22-s + 3.18e3·23-s − 768·24-s − 1.29e4·26-s + 12·27-s − 5.47e3·28-s − 2.48e3·29-s − 1.18e3·31-s + 6.14e3·32-s − 1.98e3·33-s − 512·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s − 0.879·7-s + 1.41·8-s − 0.489·9-s + 1.64·11-s − 0.288·12-s − 2.64·13-s − 1.24·14-s + 5/4·16-s − 0.0537·17-s − 0.692·18-s + 0.458·19-s + 0.169·21-s + 2.32·22-s + 1.25·23-s − 0.272·24-s − 3.74·26-s + 0.00316·27-s − 1.31·28-s − 0.547·29-s − 0.220·31-s + 1.06·32-s − 0.316·33-s − 0.0759·34-s + ⋯

Functional equation

Λ(s)=(902500s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(902500s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22541922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 23214.923214.9
Root analytic conductor: 12.343612.3436
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 902500, ( :5/2,5/2), 1)(4,\ 902500,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
5 1 1
19C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
good3D4D_{4} 1+pT+128T2+p6T3+p10T4 1 + p T + 128 T^{2} + p^{6} T^{3} + p^{10} T^{4}
7D4D_{4} 1+114T+31099T2+114p5T3+p10T4 1 + 114 T + 31099 T^{2} + 114 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1661T+430972T2661p5T3+p10T4 1 - 661 T + 430972 T^{2} - 661 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+1613T+1384022T2+1613p5T3+p10T4 1 + 1613 T + 1384022 T^{2} + 1613 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1+64T+2804713T2+64p5T3+p10T4 1 + 64 T + 2804713 T^{2} + 64 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 13185T+14543782T23185p5T3+p10T4 1 - 3185 T + 14543782 T^{2} - 3185 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+2481T+6815332T2+2481p5T3+p10T4 1 + 2481 T + 6815332 T^{2} + 2481 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+1180T+21633278T2+1180p5T3+p10T4 1 + 1180 T + 21633278 T^{2} + 1180 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1+10488T+155529814T2+10488p5T3+p10T4 1 + 10488 T + 155529814 T^{2} + 10488 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 116630T+170295586T216630p5T3+p10T4 1 - 16630 T + 170295586 T^{2} - 16630 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+11303T+297510638T2+11303p5T3+p10T4 1 + 11303 T + 297510638 T^{2} + 11303 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 112155T+47754214T212155p5T3+p10T4 1 - 12155 T + 47754214 T^{2} - 12155 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+20585T+812203882T2+20585p5T3+p10T4 1 + 20585 T + 812203882 T^{2} + 20585 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+78581T+2971672216T2+78581p5T3+p10T4 1 + 78581 T + 2971672216 T^{2} + 78581 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 143621T+1919695356T243621p5T3+p10T4 1 - 43621 T + 1919695356 T^{2} - 43621 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+7805T748756690T2+7805p5T3+p10T4 1 + 7805 T - 748756690 T^{2} + 7805 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+62488T+4005427642T2+62488p5T3+p10T4 1 + 62488 T + 4005427642 T^{2} + 62488 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+16218T+1004140843T2+16218p5T3+p10T4 1 + 16218 T + 1004140843 T^{2} + 16218 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 167122T+7273984870T267122p5T3+p10T4 1 - 67122 T + 7273984870 T^{2} - 67122 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 110714T+5751433246T210714p5T3+p10T4 1 - 10714 T + 5751433246 T^{2} - 10714 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1128188T+8689285330T2128188p5T3+p10T4 1 - 128188 T + 8689285330 T^{2} - 128188 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+178558T+22668743394T2+178558p5T3+p10T4 1 + 178558 T + 22668743394 T^{2} + 178558 p^{5} T^{3} + p^{10} T^{4}
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   L(s)=p j=14(1αj,pps)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

−9.129434144609116193537200684772, −9.030320106263371284530200187759, −7.926825121170063782183372099451, −7.69905186000330746030367434424, −7.14630541703623830423773226349, −6.89283237113185984173114475938, −6.41566966873357572970608230544, −6.18982834416182642654956277109, −5.32681871822788878088288475727, −5.29162062857896153583668196511, −4.71336109804064401555062674096, −4.28163452685650426557895937720, −3.72595559469754306097378865617, −3.20689990266173227905758790300, −2.83144276114041174353568395192, −2.39867039197355751635816936463, −1.62859656729032328239270440338, −1.16302896072277631483145537948, 0, 0, 1.16302896072277631483145537948, 1.62859656729032328239270440338, 2.39867039197355751635816936463, 2.83144276114041174353568395192, 3.20689990266173227905758790300, 3.72595559469754306097378865617, 4.28163452685650426557895937720, 4.71336109804064401555062674096, 5.29162062857896153583668196511, 5.32681871822788878088288475727, 6.18982834416182642654956277109, 6.41566966873357572970608230544, 6.89283237113185984173114475938, 7.14630541703623830423773226349, 7.69905186000330746030367434424, 7.926825121170063782183372099451, 9.030320106263371284530200187759, 9.129434144609116193537200684772

Graph of the ZZ-function along the critical line