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Properties

Label 4-4050e2-1.1-c1e2-0-15
Degree 44
Conductor 1640250016402500
Sign 11
Analytic cond. 1045.831045.83
Root an. cond. 5.686775.68677
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 6·11-s − 10·13-s − 4·14-s + 5·16-s − 2·19-s − 12·22-s + 20·26-s + 6·28-s + 6·29-s − 8·31-s − 6·32-s + 2·37-s + 4·38-s + 12·41-s − 4·43-s + 18·44-s + 12·47-s + 49-s − 30·52-s + 6·53-s − 8·56-s − 12·58-s + 18·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 1.80·11-s − 2.77·13-s − 1.06·14-s + 5/4·16-s − 0.458·19-s − 2.55·22-s + 3.92·26-s + 1.13·28-s + 1.11·29-s − 1.43·31-s − 1.06·32-s + 0.328·37-s + 0.648·38-s + 1.87·41-s − 0.609·43-s + 2.71·44-s + 1.75·47-s + 1/7·49-s − 4.16·52-s + 0.824·53-s − 1.06·56-s − 1.57·58-s + 2.34·59-s + ⋯

Functional equation

Λ(s)=(16402500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16402500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1640250016402500    =    22⋅38⋅542^{2} \cdot 3^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 1045.831045.83
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 16402500, ( :1/2,1/2), 1)(4,\ 16402500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5004527861.500452786
L(12)L(\frac12) ≈\approx 1.5004527861.500452786
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}
3 1 1
5 1 1
good7D4D_{4} 1−2T+3T2−2pT3+p2T4 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−6T+28T2−6pT3+p2T4 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+10T+48T2+10pT3+p2T4 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+31T2+p2T4 1 + 31 T^{2} + p^{2} T^{4}
19D4D_{4} 1+2T+36T2+2pT3+p2T4 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29D4D_{4} 1−6T+64T2−6pT3+p2T4 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+8T+66T2+8pT3+p2T4 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−2T+48T2−2pT3+p2T4 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−12T+91T2−12pT3+p2T4 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+4T+42T2+4pT3+p2T4 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−12T+127T2−12pT3+p2T4 1 - 12 T + 127 T^{2} - 12 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−6T+112T2−6pT3+p2T4 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−18T+196T2−18pT3+p2T4 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−2T+108T2−2pT3+p2T4 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
73D4D_{4} 1+10T+63T2+10pT3+p2T4 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4}
79C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
83D4D_{4} 1+18T+172T2+18pT3+p2T4 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−12T+187T2−12pT3+p2T4 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−2T+147T2−2pT3+p2T4 1 - 2 T + 147 T^{2} - 2 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

−8.585961035340322910504969741762, −8.564351939549586200275490031057, −7.63378963683121853855758676117, −7.60553535900320545667296771853, −7.24133422572765208007956838028, −7.16243791595589116813123827045, −6.46886008021186023190180983011, −6.26717189261945367055548223165, −5.77997415909167863345592406906, −5.31642141141855173206294673717, −4.79132564062844985577578188614, −4.59348328407234204001486845554, −3.92196839120839761454897008847, −3.74738704736386788635380242209, −2.77204138034596061285246910137, −2.62505839480781160542173399591, −1.96826623906596701698077805252, −1.81182661570116069200964044476, −0.899992437945576909093392677124, −0.55819194998023650950123426304, 0.55819194998023650950123426304, 0.899992437945576909093392677124, 1.81182661570116069200964044476, 1.96826623906596701698077805252, 2.62505839480781160542173399591, 2.77204138034596061285246910137, 3.74738704736386788635380242209, 3.92196839120839761454897008847, 4.59348328407234204001486845554, 4.79132564062844985577578188614, 5.31642141141855173206294673717, 5.77997415909167863345592406906, 6.26717189261945367055548223165, 6.46886008021186023190180983011, 7.16243791595589116813123827045, 7.24133422572765208007956838028, 7.60553535900320545667296771853, 7.63378963683121853855758676117, 8.564351939549586200275490031057, 8.585961035340322910504969741762

Graph of the ZZ-function along the critical line