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Properties

Label 4-3822e2-1.1-c1e2-0-10
Degree 44
Conductor 1460768414607684
Sign 11
Analytic cond. 931.398931.398
Root an. cond. 5.524385.52438
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s − 4·11-s + 6·12-s + 2·13-s − 4·15-s + 5·16-s − 8·17-s − 6·18-s + 10·19-s − 6·20-s + 8·22-s − 10·23-s − 8·24-s − 4·26-s + 4·27-s − 2·29-s + 8·30-s − 4·31-s − 6·32-s − 8·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s − 1.20·11-s + 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 2.29·19-s − 1.34·20-s + 1.70·22-s − 2.08·23-s − 1.63·24-s − 0.784·26-s + 0.769·27-s − 0.371·29-s + 1.46·30-s − 0.718·31-s − 1.06·32-s − 1.39·33-s + ⋯

Functional equation

Λ(s)=(14607684s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(14607684s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1460768414607684    =    22⋅32⋅74⋅1322^{2} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 931.398931.398
Root analytic conductor: 5.524385.52438
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 14607684, ( :1/2,1/2), 1)(4,\ 14607684,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
3C1C_1 (1−T)2 ( 1 - T )^{2}
7 1 1
13C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+2T+4T2+2pT3+p2T4 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+4T+19T2+4pT3+p2T4 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+8T+43T2+8pT3+p2T4 1 + 8 T + 43 T^{2} + 8 p T^{3} + p^{2} T^{4}
19C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
23D4D_{4} 1+10T+64T2+10pT3+p2T4 1 + 10 T + 64 T^{2} + 10 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T+31T2+2pT3+p2T4 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+4T+38T2+4pT3+p2T4 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−6T+76T2−6pT3+p2T4 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+10T+100T2+10pT3+p2T4 1 + 10 T + 100 T^{2} + 10 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
47C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
53C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
59C22C_2^2 1+55T2+p2T4 1 + 55 T^{2} + p^{2} T^{4}
61D4D_{4} 1+8T+75T2+8pT3+p2T4 1 + 8 T + 75 T^{2} + 8 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+6T+31T2+6pT3+p2T4 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−22T+235T2−22pT3+p2T4 1 - 22 T + 235 T^{2} - 22 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+22T+260T2+22pT3+p2T4 1 + 22 T + 260 T^{2} + 22 p T^{3} + p^{2} T^{4}
79C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
83D4D_{4} 1+16T+202T2+16pT3+p2T4 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−18T+196T2−18pT3+p2T4 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+14T+180T2+14pT3+p2T4 1 + 14 T + 180 T^{2} + 14 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

−8.137268305236160172415194446149, −8.075866292435197277483842580689, −7.62761906499602603133425477889, −7.61723073235035246849060056607, −6.91843924780574986112825004724, −6.89985659807353539749792378616, −6.24092644997960384654825484801, −5.70378190625531471709317989893, −5.51237617004546589237254956116, −4.83677546507183315512320937418, −4.17317481185997228729501704676, −4.14316911150518593870444560600, −3.36099736752790372719097445675, −3.18906377798137353750742715907, −2.56525921700966039264616764806, −2.28330578840492803100587690073, −1.63302423646835084744839110438, −1.24172395683391563791530561812, 0, 0, 1.24172395683391563791530561812, 1.63302423646835084744839110438, 2.28330578840492803100587690073, 2.56525921700966039264616764806, 3.18906377798137353750742715907, 3.36099736752790372719097445675, 4.14316911150518593870444560600, 4.17317481185997228729501704676, 4.83677546507183315512320937418, 5.51237617004546589237254956116, 5.70378190625531471709317989893, 6.24092644997960384654825484801, 6.89985659807353539749792378616, 6.91843924780574986112825004724, 7.61723073235035246849060056607, 7.62761906499602603133425477889, 8.075866292435197277483842580689, 8.137268305236160172415194446149

Graph of the ZZ-function along the critical line