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Properties

Label 4-3e10-1.1-c1e2-0-0
Degree 44
Conductor 5904959049
Sign 11
Analytic cond. 3.765013.76501
Root an. cond. 1.392961.39296
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 − 2·7-s + 10·13-s − 3·16-s − 2·19-s + 2·25-s + 2·28-s + 10·31-s − 2·37-s − 2·43-s − 11·49-s − 10·52-s + 4·61-s + 7·64-s + 16·67-s + 4·73-s + 2·76-s − 2·79-s − 20·91-s + 34·97-s − 2·100-s + 16·103-s + 34·109-s + 6·112-s − 10·121-s − 10·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s + 2.77·13-s − 3/4·16-s − 0.458·19-s + 2/5·25-s + 0.377·28-s + 1.79·31-s − 0.328·37-s − 0.304·43-s − 1.57·49-s − 1.38·52-s + 0.512·61-s + 7/8·64-s + 1.95·67-s + 0.468·73-s + 0.229·76-s − 0.225·79-s − 2.09·91-s + 3.45·97-s − 1/5·100-s + 1.57·103-s + 3.25·109-s + 0.566·112-s − 0.909·121-s − 0.898·124-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=(59049s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(59049s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5904959049    =    3103^{10}
Sign: 11
Analytic conductor: 3.765013.76501
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 59049, ( :1/2,1/2), 1)(4,\ 59049,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2139641381.213964138
L(12)L(\frac12) ≈\approx 1.2139641381.213964138
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)
bad3 1 1
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
37C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
53C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
59C22C_2^2 1+106T2+p2T4 1 + 106 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
73C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
79C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
83C22C_2^2 1+118T2+p2T4 1 + 118 T^{2} + p^{2} T^{4}
89C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
97C2C_2 (1−17T+pT2)2 ( 1 - 17 T + p T^{2} )^{2}
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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

−12.60055799547426009652810274468, −11.68613044632300397194764208501, −11.27363637433942366154952747032, −11.14495484165665719309580933705, −10.21723916378989491163277649923, −10.13654896803195556492580659962, −9.388168687038182801682601425010, −8.833245131673608769163171550514, −8.519652151235185896572998162213, −8.223590322904053967862384369796, −7.37225555795780709109469246797, −6.51667717431085117725053286253, −6.38988047348155697162740997975, −5.94367436151325889775645239570, −4.95763268889965373286462212722, −4.50551446413152782472606775083, −3.52168957571220695631214088920, −3.47172281941875945571553746616, −2.21095194874431803639976263612, −0.970090806857385628640518965887, 0.970090806857385628640518965887, 2.21095194874431803639976263612, 3.47172281941875945571553746616, 3.52168957571220695631214088920, 4.50551446413152782472606775083, 4.95763268889965373286462212722, 5.94367436151325889775645239570, 6.38988047348155697162740997975, 6.51667717431085117725053286253, 7.37225555795780709109469246797, 8.223590322904053967862384369796, 8.519652151235185896572998162213, 8.833245131673608769163171550514, 9.388168687038182801682601425010, 10.13654896803195556492580659962, 10.21723916378989491163277649923, 11.14495484165665719309580933705, 11.27363637433942366154952747032, 11.68613044632300397194764208501, 12.60055799547426009652810274468

Graph of the ZZ-function along the critical line