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Properties

Label 4-105e2-1.1-c1e2-0-8
Degree 44
Conductor 1102511025
Sign 11
Analytic cond. 0.7029630.702963
Root an. cond. 0.9156570.915657
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  + 3·4-s + 2·5-s − 9-s − 12·11-s + 5·16-s + 12·19-s + 6·20-s − 25-s + 4·29-s − 20·31-s − 3·36-s + 4·41-s − 36·44-s − 2·45-s − 49-s − 24·55-s + 16·59-s − 4·61-s + 3·64-s + 20·71-s + 36·76-s − 8·79-s + 10·80-s + 81-s − 12·89-s + 24·95-s + 12·99-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s − 1/3·9-s − 3.61·11-s + 5/4·16-s + 2.75·19-s + 1.34·20-s − 1/5·25-s + 0.742·29-s − 3.59·31-s − 1/2·36-s + 0.624·41-s − 5.42·44-s − 0.298·45-s − 1/7·49-s − 3.23·55-s + 2.08·59-s − 0.512·61-s + 3/8·64-s + 2.37·71-s + 4.12·76-s − 0.900·79-s + 1.11·80-s + 1/9·81-s − 1.27·89-s + 2.46·95-s + 1.20·99-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.7029630.702963
Root analytic conductor: 0.9156570.915657
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11025, ( :1/2,1/2), 1)(4,\ 11025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3483404871.348340487
L(12)L(\frac12) ≈\approx 1.3483404871.348340487
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)
bad3C2C_2 1+T2 1 + T^{2}
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
7C2C_2 1+T2 1 + T^{2}
good2C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
11C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
19C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
23C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
37C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C22C_2^2 1+122T2+p2T4 1 + 122 T^{2} + p^{2} T^{4}
71C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1−190T2+p2T4 1 - 190 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

−14.07765741036902913605419866632, −13.34777610896378407247861108327, −13.03878561228530477495552939210, −12.59576519015341727507158702442, −11.86948020238545760806007327417, −11.27433118061208772548131650751, −10.69192697832386041230196696032, −10.65843719583477371813672800041, −9.704083858777687059690286950373, −9.590614160303390257499736002489, −8.370119229283450604227174471651, −7.77449242194542902577294645997, −7.40285334522283855310331904915, −6.96295836787847011215570227142, −5.66673716415926716727854405042, −5.53092384774540902824306223888, −5.20934549967759721253207672094, −3.31926418578659151487141795523, −2.70173620439600678454391775595, −2.06135419027838012408223731157, 2.06135419027838012408223731157, 2.70173620439600678454391775595, 3.31926418578659151487141795523, 5.20934549967759721253207672094, 5.53092384774540902824306223888, 5.66673716415926716727854405042, 6.96295836787847011215570227142, 7.40285334522283855310331904915, 7.77449242194542902577294645997, 8.370119229283450604227174471651, 9.590614160303390257499736002489, 9.704083858777687059690286950373, 10.65843719583477371813672800041, 10.69192697832386041230196696032, 11.27433118061208772548131650751, 11.86948020238545760806007327417, 12.59576519015341727507158702442, 13.03878561228530477495552939210, 13.34777610896378407247861108327, 14.07765741036902913605419866632

Graph of the ZZ-function along the critical line