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Properties

Label 4-7644e2-1.1-c1e2-0-9
Degree 44
Conductor 5843073658430736
Sign 11
Analytic cond. 3725.593725.59
Root an. cond. 7.812657.81265
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  − 2·3-s − 2·5-s + 3·9-s + 2·11-s + 2·13-s + 4·15-s + 4·17-s + 4·19-s + 8·23-s − 4·25-s − 4·27-s + 8·31-s − 4·33-s + 8·37-s − 4·39-s − 6·41-s − 8·43-s − 6·45-s + 6·47-s − 8·51-s + 20·53-s − 4·55-s − 8·57-s − 26·59-s − 4·61-s − 4·65-s + 20·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.769·27-s + 1.43·31-s − 0.696·33-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.894·45-s + 0.875·47-s − 1.12·51-s + 2.74·53-s − 0.539·55-s − 1.05·57-s − 3.38·59-s − 0.512·61-s − 0.496·65-s + 2.44·67-s + ⋯

Functional equation

Λ(s)=(58430736s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(58430736s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5843073658430736    =    24⋅32⋅74⋅1322^{4} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 3725.593725.59
Root analytic conductor: 7.812657.81265
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 58430736, ( :1/2,1/2), 1)(4,\ 58430736,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2851131172.285113117
L(12)L(\frac12) ≈\approx 2.2851131172.285113117
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)
bad2 1 1
3C1C_1 (1+T)2 ( 1 + T )^{2}
7 1 1
13C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+2T+8T2+2pT3+p2T4 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−2T−4T2−2pT3+p2T4 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−4T+26T2−4pT3+p2T4 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−8T+50T2−8pT3+p2T4 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
37D4D_{4} 1−8T+78T2−8pT3+p2T4 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+88T2+6pT3+p2T4 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47D4D_{4} 1−6T+100T2−6pT3+p2T4 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4}
53C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
59D4D_{4} 1+26T+284T2+26pT3+p2T4 1 + 26 T + 284 T^{2} + 26 p T^{3} + p^{2} T^{4}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67D4D_{4} 1−20T+222T2−20pT3+p2T4 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−14T+116T2−14pT3+p2T4 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+8T+54T2+8pT3+p2T4 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+162T2−8pT3+p2T4 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−2T+20T2−2pT3+p2T4 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+6T+160T2+6pT3+p2T4 1 + 6 T + 160 T^{2} + 6 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + 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

−7.904696133387382920601185164506, −7.77152843674603387036747726129, −7.16700630810904931642192852650, −7.10452097416777523923980011322, −6.58396165623470241685486679285, −6.41777312565683030939578173483, −5.82412222586304801209055193485, −5.74062457578461743406329773023, −5.12884698935276346411162174747, −5.03078856656000555328252644077, −4.51679823689375048624717539146, −4.16100237133683056497631379124, −3.72352403892414804948529716609, −3.51310115297367884254414380209, −2.95170243566927728413181865760, −2.63205757969519888375762801463, −1.78179794534730876657303852285, −1.35006073233233589063754025141, −0.802092637331039509163911519313, −0.57852670315661870629138675603, 0.57852670315661870629138675603, 0.802092637331039509163911519313, 1.35006073233233589063754025141, 1.78179794534730876657303852285, 2.63205757969519888375762801463, 2.95170243566927728413181865760, 3.51310115297367884254414380209, 3.72352403892414804948529716609, 4.16100237133683056497631379124, 4.51679823689375048624717539146, 5.03078856656000555328252644077, 5.12884698935276346411162174747, 5.74062457578461743406329773023, 5.82412222586304801209055193485, 6.41777312565683030939578173483, 6.58396165623470241685486679285, 7.10452097416777523923980011322, 7.16700630810904931642192852650, 7.77152843674603387036747726129, 7.904696133387382920601185164506

Graph of the ZZ-function along the critical line