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Properties

Label 4-704e2-1.1-c1e2-0-17
Degree 44
Conductor 495616495616
Sign 11
Analytic cond. 31.600931.6009
Root an. cond. 2.370962.37096
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  − 3-s − 3·5-s − 9-s − 2·11-s − 4·13-s + 3·15-s + 6·17-s − 6·19-s − 3·23-s + 25-s − 6·29-s − 15·31-s + 2·33-s − 37-s + 4·39-s − 6·43-s + 3·45-s − 8·47-s − 14·49-s − 6·51-s + 6·55-s + 6·57-s + 13·59-s + 6·61-s + 12·65-s − 3·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.774·15-s + 1.45·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s − 2.69·31-s + 0.348·33-s − 0.164·37-s + 0.640·39-s − 0.914·43-s + 0.447·45-s − 1.16·47-s − 2·49-s − 0.840·51-s + 0.809·55-s + 0.794·57-s + 1.69·59-s + 0.768·61-s + 1.48·65-s − 0.366·67-s + 0.361·69-s + ⋯

Functional equation

Λ(s)=(495616s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(495616s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 495616495616    =    212⋅1122^{12} \cdot 11^{2}
Sign: 11
Analytic conductor: 31.600931.6009
Root analytic conductor: 2.370962.37096
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 495616, ( :1/2,1/2), 1)(4,\ 495616,\ (\ :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
11C1C_1 (1+T)2 ( 1 + T )^{2}
good3D4D_{4} 1+T+2T2+pT3+p2T4 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}
5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17D4D_{4} 1−6T+26T2−6pT3+p2T4 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+6T+30T2+6pT3+p2T4 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+3T+10T2+3pT3+p2T4 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+6T+50T2+6pT3+p2T4 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+15T+114T2+15pT3+p2T4 1 + 15 T + 114 T^{2} + 15 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+T+36T2+pT3+p2T4 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4}
41C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
43D4D_{4} 1+6T+78T2+6pT3+p2T4 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4}
47C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59D4D_{4} 1−13T+122T2−13pT3+p2T4 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−6T−22T2−6pT3+p2T4 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+3T+98T2+3pT3+p2T4 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−19T+194T2−19pT3+p2T4 1 - 19 T + 194 T^{2} - 19 p T^{3} + p^{2} T^{4}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79D4D_{4} 1+18T+222T2+18pT3+p2T4 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+2T+14T2+2pT3+p2T4 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+3T+176T2+3pT3+p2T4 1 + 3 T + 176 T^{2} + 3 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+T+156T2+pT3+p2T4 1 + T + 156 T^{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

−9.994066662245861355755360111118, −9.965088507638647073968732990367, −9.623515760419739714697837517073, −8.656929244624234630824060750928, −8.530945589208049100672885995963, −8.011978207178670625658766419097, −7.49674853250841138701759634638, −7.35414587092176273137964814482, −6.84666074039592107350230004766, −6.09650404237675435814866983305, −5.65703739115574662441780954724, −5.28888958868363812420471932101, −4.78554179254037215939622947776, −4.17676354055744813843210672677, −3.48678432040528483062222868330, −3.41502641256963178080103047658, −2.30906917918481945396602919838, −1.72303323645901048412833725201, 0, 0, 1.72303323645901048412833725201, 2.30906917918481945396602919838, 3.41502641256963178080103047658, 3.48678432040528483062222868330, 4.17676354055744813843210672677, 4.78554179254037215939622947776, 5.28888958868363812420471932101, 5.65703739115574662441780954724, 6.09650404237675435814866983305, 6.84666074039592107350230004766, 7.35414587092176273137964814482, 7.49674853250841138701759634638, 8.011978207178670625658766419097, 8.530945589208049100672885995963, 8.656929244624234630824060750928, 9.623515760419739714697837517073, 9.965088507638647073968732990367, 9.994066662245861355755360111118

Graph of the ZZ-function along the critical line