Properties

Label 4-234e2-1.1-c5e2-0-1
Degree 44
Conductor 5475654756
Sign 11
Analytic cond. 1408.481408.48
Root an. cond. 6.126156.12615
Motivic weight 55
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  − 16·4-s − 1.19e3·13-s + 256·16-s + 2.20e3·17-s − 2.10e3·23-s + 3.64e3·25-s + 8.20e3·29-s + 1.99e4·43-s + 2.25e4·49-s + 1.91e4·52-s + 1.50e3·53-s − 1.15e5·61-s − 4.09e3·64-s − 3.52e4·68-s + 1.26e5·79-s + 3.36e4·92-s − 5.83e4·100-s + 2.26e5·101-s + 5.00e4·103-s − 4.98e4·107-s − 2.00e5·113-s − 1.31e5·116-s + 3.07e5·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.96·13-s + 1/4·16-s + 1.84·17-s − 0.827·23-s + 1.16·25-s + 1.81·29-s + 1.64·43-s + 1.34·49-s + 0.981·52-s + 0.0733·53-s − 3.98·61-s − 1/8·64-s − 0.923·68-s + 2.27·79-s + 0.413·92-s − 0.583·100-s + 2.20·101-s + 0.465·103-s − 0.420·107-s − 1.47·113-s − 0.906·116-s + 1.91·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

Functional equation

Λ(s)=(54756s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(54756s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5475654756    =    22341322^{2} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 1408.481408.48
Root analytic conductor: 6.126156.12615
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 54756, ( :5/2,5/2), 1)(4,\ 54756,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) \approx 2.1816780032.181678003
L(12)L(\frac12) \approx 2.1816780032.181678003
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)
bad2C2C_2 1+p4T2 1 + p^{4} T^{2}
3 1 1
13C2C_2 1+92pT+p5T2 1 + 92 p T + p^{5} T^{2}
good5C22C_2^2 13649T2+p10T4 1 - 3649 T^{2} + p^{10} T^{4}
7C22C_2^2 1461p2T2+p10T4 1 - 461 p^{2} T^{2} + p^{10} T^{4}
11C22C_2^2 1307702T2+p10T4 1 - 307702 T^{2} + p^{10} T^{4}
17C2C_2 (11101T+p5T2)2 ( 1 - 1101 T + p^{5} T^{2} )^{2}
19C22C_2^2 13583298T2+p10T4 1 - 3583298 T^{2} + p^{10} T^{4}
23C2C_2 (1+1050T+p5T2)2 ( 1 + 1050 T + p^{5} T^{2} )^{2}
29C2C_2 (14104T+p5T2)2 ( 1 - 4104 T + p^{5} T^{2} )^{2}
31C22C_2^2 1+35363074T2+p10T4 1 + 35363074 T^{2} + p^{10} T^{4}
37C22C_2^2 162841233T2+p10T4 1 - 62841233 T^{2} + p^{10} T^{4}
41C22C_2^2 1141842002T2+p10T4 1 - 141842002 T^{2} + p^{10} T^{4}
43C2C_2 (19995T+p5T2)2 ( 1 - 9995 T + p^{5} T^{2} )^{2}
47C22C_2^2 1450028765T2+p10T4 1 - 450028765 T^{2} + p^{10} T^{4}
53C2C_2 (1750T+p5T2)2 ( 1 - 750 T + p^{5} T^{2} )^{2}
59C22C_2^2 1+246071246T2+p10T4 1 + 246071246 T^{2} + p^{10} T^{4}
61C2C_2 (1+57920T+p5T2)2 ( 1 + 57920 T + p^{5} T^{2} )^{2}
67C22C_2^2 12179862870T2+p10T4 1 - 2179862870 T^{2} + p^{10} T^{4}
71C22C_2^2 1+454456379T2+p10T4 1 + 454456379 T^{2} + p^{10} T^{4}
73C22C_2^2 1680937230T2+p10T4 1 - 680937230 T^{2} + p^{10} T^{4}
79C2C_2 (163202T+p5T2)2 ( 1 - 63202 T + p^{5} T^{2} )^{2}
83C22C_2^2 14802491522T2+p10T4 1 - 4802491522 T^{2} + p^{10} T^{4}
89C22C_2^2 1189689614T2+p10T4 1 - 189689614 T^{2} + p^{10} T^{4}
97C22C_2^2 1+8570163790T2+p10T4 1 + 8570163790 T^{2} + 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

−12.01564722989896368056223299065, −10.85941592912127044059202774695, −10.41389127526647543608719727596, −10.23679384610399907726125006126, −9.502633645548166488651127554080, −9.302176830956230348269183314698, −8.662558588919773944038673621331, −7.911105876696710420724617482656, −7.69034022700876328380984609525, −7.20402634972305475145049259457, −6.46717429904714964533298894295, −5.84144785520744976759359268610, −5.31584219160445818996791528294, −4.63755667528941151464541650240, −4.41321701233527808421709477405, −3.33124898745015316029784352709, −2.88290570232400841982222788082, −2.14130213880402589254586535111, −1.07623588320045590845175062375, −0.50840447317880579821769245286, 0.50840447317880579821769245286, 1.07623588320045590845175062375, 2.14130213880402589254586535111, 2.88290570232400841982222788082, 3.33124898745015316029784352709, 4.41321701233527808421709477405, 4.63755667528941151464541650240, 5.31584219160445818996791528294, 5.84144785520744976759359268610, 6.46717429904714964533298894295, 7.20402634972305475145049259457, 7.69034022700876328380984609525, 7.911105876696710420724617482656, 8.662558588919773944038673621331, 9.302176830956230348269183314698, 9.502633645548166488651127554080, 10.23679384610399907726125006126, 10.41389127526647543608719727596, 10.85941592912127044059202774695, 12.01564722989896368056223299065

Graph of the ZZ-function along the critical line