Properties

Label 4-936e2-1.1-c1e2-0-28
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·7-s + 6·19-s + 2·25-s + 6·31-s − 4·37-s + 8·43-s + 14·49-s − 20·61-s + 2·67-s + 4·79-s + 8·103-s − 8·109-s − 10·121-s + 127-s + 131-s + 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 12·175-s + 179-s + ⋯
L(s)  = 1  + 2.26·7-s + 1.37·19-s + 2/5·25-s + 1.07·31-s − 0.657·37-s + 1.21·43-s + 2·49-s − 2.56·61-s + 0.244·67-s + 0.450·79-s + 0.788·103-s − 0.766·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + 0.907·175-s + 0.0747·179-s + ⋯

Functional equation

Λ(s)=(876096s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(876096s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 876096876096    =    26341322^{6} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 876096, ( :1/2,1/2), 1)(4,\ 876096,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) \approx 3.1202659923.120265992
L(12)L(\frac12) \approx 3.1202659923.120265992
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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
13C1C_1×\timesC1C_1 (1T)(1+T) ( 1 - T )( 1 + T )
good5C22C_2^2 12T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2×\timesC2C_2 (14T+pT2)(12T+pT2) ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
17C2C_2 (14T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C2C_2×\timesC2C_2 (14T+pT2)(12T+pT2) ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )
23C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
29C22C_2^2 138T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (110T+pT2)(1+4T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2×\timesC2C_2 (14T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43C2C_2 (14T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
53C22C_2^2 170T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (112T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2×\timesC2C_2 (12T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
71C22C_2^2 166T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
73C2C_2 (14T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
79C2C_2×\timesC2C_2 (18T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1+118T2+p2T4 1 + 118 T^{2} + p^{2} T^{4}
89C2C_2 (118T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )
97C2C_2 (116T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
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   L(s)=p j=14(1αj,pps)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

−8.075257102118721441562989676744, −7.86756139395375305198988393851, −7.41778155909519335240701448999, −7.04107829495058172241014800754, −6.40290829742781820907724539550, −5.84122750036013329338006700633, −5.38931768486352998317478144494, −4.97504874090531212686413887179, −4.55334630045497318040636110189, −4.25197876941557033379623437614, −3.37613329269396856069104097184, −2.89486831361438852855982285933, −2.10529677189616315902088168255, −1.52669754955306856650657171662, −0.934741245515398847185651189012, 0.934741245515398847185651189012, 1.52669754955306856650657171662, 2.10529677189616315902088168255, 2.89486831361438852855982285933, 3.37613329269396856069104097184, 4.25197876941557033379623437614, 4.55334630045497318040636110189, 4.97504874090531212686413887179, 5.38931768486352998317478144494, 5.84122750036013329338006700633, 6.40290829742781820907724539550, 7.04107829495058172241014800754, 7.41778155909519335240701448999, 7.86756139395375305198988393851, 8.075257102118721441562989676744

Graph of the ZZ-function along the critical line