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Properties

Label 4-9000e2-1.1-c1e2-0-11
Degree 44
Conductor 8100000081000000
Sign 11
Analytic cond. 5164.635164.63
Root an. cond. 8.477348.47734
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s − 6·13-s + 9·17-s − 3·19-s − 23-s − 2·29-s − 9·31-s − 2·37-s + 2·41-s − 8·43-s + 5·47-s − 14·49-s + 11·53-s + 2·59-s + 61-s − 12·67-s − 6·71-s − 11·79-s + 17·83-s − 18·89-s − 4·97-s − 24·101-s − 2·103-s + 107-s + 21·109-s + 113-s − 14·121-s + ⋯
L(s)  = 1  + 0.603·11-s − 1.66·13-s + 2.18·17-s − 0.688·19-s − 0.208·23-s − 0.371·29-s − 1.61·31-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 0.729·47-s − 2·49-s + 1.51·53-s + 0.260·59-s + 0.128·61-s − 1.46·67-s − 0.712·71-s − 1.23·79-s + 1.86·83-s − 1.90·89-s − 0.406·97-s − 2.38·101-s − 0.197·103-s + 0.0966·107-s + 2.01·109-s + 0.0940·113-s − 1.27·121-s + ⋯

Functional equation

Λ(s)=(81000000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(81000000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8100000081000000    =    26⋅34⋅562^{6} \cdot 3^{4} \cdot 5^{6}
Sign: 11
Analytic conductor: 5164.635164.63
Root analytic conductor: 8.477348.47734
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 81000000, ( :1/2,1/2), 1)(4,\ 81000000,\ (\ :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)
bad2 1 1
3 1 1
5 1 1
good7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11D4D_{4} 1−2T+18T2−2pT3+p2T4 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+6T+30T2+6pT3+p2T4 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−9T+43T2−9pT3+p2T4 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+3T+39T2+3pT3+p2T4 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+T+35T2+pT3+p2T4 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T+54T2+2pT3+p2T4 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+9T+81T2+9pT3+p2T4 1 + 9 T + 81 T^{2} + 9 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+2T+70T2+2pT3+p2T4 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−2T+78T2−2pT3+p2T4 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+8T+82T2+8pT3+p2T4 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−5T+69T2−5pT3+p2T4 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−11T+135T2−11pT3+p2T4 1 - 11 T + 135 T^{2} - 11 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−2T−6T2−2pT3+p2T4 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−T+121T2−pT3+p2T4 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4}
67D4D_{4} 1+12T+90T2+12pT3+p2T4 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4}
71C4C_4 1+6T+26T2+6pT3+p2T4 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+66T2+p2T4 1 + 66 T^{2} + p^{2} T^{4}
79D4D_{4} 1+11T+187T2+11pT3+p2T4 1 + 11 T + 187 T^{2} + 11 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−17T+177T2−17pT3+p2T4 1 - 17 T + 177 T^{2} - 17 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+18T+254T2+18pT3+p2T4 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+4T+118T2+4pT3+p2T4 1 + 4 T + 118 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

−7.52717325159768530270969810696, −7.28803927068176901567528403245, −6.85840273628066243864076422096, −6.75747143394682731722574367243, −6.09828004558912712089416354179, −5.79271710041246260187830179398, −5.47023078486040518970966375597, −5.29408613760418294991650863625, −4.63128155981621757644337285054, −4.62843278652524055023556625283, −3.85858639540731446301337743784, −3.76529344533187780122124035172, −3.20411575180936709261441894958, −2.96959269291306325093127058862, −2.30346121974547957596504213333, −2.07410347801584169966326927247, −1.33166903669837875939490112405, −1.21521918029060498041211452602, 0, 0, 1.21521918029060498041211452602, 1.33166903669837875939490112405, 2.07410347801584169966326927247, 2.30346121974547957596504213333, 2.96959269291306325093127058862, 3.20411575180936709261441894958, 3.76529344533187780122124035172, 3.85858639540731446301337743784, 4.62843278652524055023556625283, 4.63128155981621757644337285054, 5.29408613760418294991650863625, 5.47023078486040518970966375597, 5.79271710041246260187830179398, 6.09828004558912712089416354179, 6.75747143394682731722574367243, 6.85840273628066243864076422096, 7.28803927068176901567528403245, 7.52717325159768530270969810696

Graph of the ZZ-function along the critical line