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Properties

Label 4-1110e2-1.1-c1e2-0-2
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
Motivic weight 11
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  − 4-s − 2·5-s − 9-s + 2·11-s + 16-s − 16·19-s + 2·20-s − 25-s + 10·29-s + 6·31-s + 36-s − 10·41-s − 2·44-s + 2·45-s − 11·49-s − 4·55-s + 8·59-s − 18·61-s − 64-s + 12·71-s + 16·76-s + 16·79-s − 2·80-s + 81-s + 8·89-s + 32·95-s − 2·99-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1/3·9-s + 0.603·11-s + 1/4·16-s − 3.67·19-s + 0.447·20-s − 1/5·25-s + 1.85·29-s + 1.07·31-s + 1/6·36-s − 1.56·41-s − 0.301·44-s + 0.298·45-s − 1.57·49-s − 0.539·55-s + 1.04·59-s − 2.30·61-s − 1/8·64-s + 1.42·71-s + 1.83·76-s + 1.80·79-s − 0.223·80-s + 1/9·81-s + 0.847·89-s + 3.28·95-s − 0.201·99-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1232100, ( :1/2,1/2), 1)(4,\ 1232100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.57750301280.5775030128
L(12)L(\frac12) ≈\approx 0.57750301280.5775030128
L(32)L(\frac{3}{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)
bad2C2C_2 1+T2 1 + T^{2}
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
37C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
11C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
23C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
29C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
31C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
43C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C22C_2^2 1−81T2+p2T4 1 - 81 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
97C22C_2^2 1−113T2+p2T4 1 - 113 T^{2} + p^{2} 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

−10.53979398143835449083076821274, −9.395669002911796927189781335534, −9.357892467028456114938748667481, −8.570350274219051471220173676509, −8.327331849187970760981990550251, −8.246948449039429908119213937451, −7.84041266157207356084912538028, −6.99906153159766326917334432871, −6.52367819123424428092002382614, −6.43622346198207719935651175720, −6.05137938684773561815803510505, −5.09486671455846120428412360008, −4.80164103970649508307300584206, −4.34645260128868524479721101036, −3.96523033989564237250482058195, −3.53273829798565745141436587912, −2.78697484479330659016921360453, −2.23110081171226043300561723713, −1.46048655681344617088755378236, −0.34266992419124572471659020251, 0.34266992419124572471659020251, 1.46048655681344617088755378236, 2.23110081171226043300561723713, 2.78697484479330659016921360453, 3.53273829798565745141436587912, 3.96523033989564237250482058195, 4.34645260128868524479721101036, 4.80164103970649508307300584206, 5.09486671455846120428412360008, 6.05137938684773561815803510505, 6.43622346198207719935651175720, 6.52367819123424428092002382614, 6.99906153159766326917334432871, 7.84041266157207356084912538028, 8.246948449039429908119213937451, 8.327331849187970760981990550251, 8.570350274219051471220173676509, 9.357892467028456114938748667481, 9.395669002911796927189781335534, 10.53979398143835449083076821274

Graph of the ZZ-function along the critical line