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Properties

Label 4-9702e2-1.1-c1e2-0-29
Degree 44
Conductor 9412880494128804
Sign 11
Analytic cond. 6001.736001.73
Root an. cond. 8.801758.80175
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 + 3·4-s + 4·8-s − 2·11-s − 8·13-s + 5·16-s + 6·17-s − 4·22-s − 3·25-s − 16·26-s − 4·29-s − 8·31-s + 6·32-s + 12·34-s − 8·37-s − 18·41-s + 8·43-s − 6·44-s − 8·47-s − 6·50-s − 24·52-s − 8·53-s − 8·58-s + 8·59-s + 8·61-s − 16·62-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s − 2.21·13-s + 5/4·16-s + 1.45·17-s − 0.852·22-s − 3/5·25-s − 3.13·26-s − 0.742·29-s − 1.43·31-s + 1.06·32-s + 2.05·34-s − 1.31·37-s − 2.81·41-s + 1.21·43-s − 0.904·44-s − 1.16·47-s − 0.848·50-s − 3.32·52-s − 1.09·53-s − 1.05·58-s + 1.04·59-s + 1.02·61-s − 2.03·62-s + 7/8·64-s + ⋯

Functional equation

Λ(s)=(94128804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(94128804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9412880494128804    =    22⋅34⋅74⋅1122^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 6001.736001.73
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 94128804, ( :1/2,1/2), 1)(4,\ 94128804,\ (\ :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}
3 1 1
7 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
good5C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
19C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
23C22C_2^2 1+39T2+p2T4 1 + 39 T^{2} + p^{2} T^{4}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37D4D_{4} 1+8T+62T2+8pT3+p2T4 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4}
41C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
43D4D_{4} 1−8T+74T2−8pT3+p2T4 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+8T+pT2+8pT3+p2T4 1 + 8 T + p T^{2} + 8 p T^{3} + p^{2} T^{4}
53C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
59D4D_{4} 1−8T+22T2−8pT3+p2T4 1 - 8 T + 22 T^{2} - 8 p T^{3} + 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} 1−16T+178T2−16pT3+p2T4 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+20T+218T2+20pT3+p2T4 1 + 20 T + 218 T^{2} + 20 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−16T+215T2−16pT3+p2T4 1 - 16 T + 215 T^{2} - 16 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+10T+79T2+10pT3+p2T4 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+16T+214T2+16pT3+p2T4 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+2T+83T2+2pT3+p2T4 1 + 2 T + 83 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

−7.29259832148892851819981694914, −7.10526583277237411484371916517, −6.89472688844500263623394192790, −6.57701248164210012835587204339, −5.85317841156047568771201320891, −5.68183884219909896951411680855, −5.28343170654185137238223581323, −5.25832770121441114498776103726, −4.79588018584106298209245175258, −4.53232244814853763024583822035, −3.80923680177797962015715438576, −3.78526277519252135488512400289, −3.18004995943381606656949025677, −3.03379175806038509593832848952, −2.46507451911545987819100337845, −2.07583230112106834888351709169, −1.73059376129159953175002230519, −1.22940663671762592696668762357, 0, 0, 1.22940663671762592696668762357, 1.73059376129159953175002230519, 2.07583230112106834888351709169, 2.46507451911545987819100337845, 3.03379175806038509593832848952, 3.18004995943381606656949025677, 3.78526277519252135488512400289, 3.80923680177797962015715438576, 4.53232244814853763024583822035, 4.79588018584106298209245175258, 5.25832770121441114498776103726, 5.28343170654185137238223581323, 5.68183884219909896951411680855, 5.85317841156047568771201320891, 6.57701248164210012835587204339, 6.89472688844500263623394192790, 7.10526583277237411484371916517, 7.29259832148892851819981694914

Graph of the ZZ-function along the critical line