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Properties

Label 4-9576e2-1.1-c1e2-0-1
Degree 44
Conductor 9169977691699776
Sign 11
Analytic cond. 5846.855846.85
Root an. cond. 8.744418.74441
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·7-s + 4·11-s + 2·13-s + 2·17-s − 2·19-s − 6·23-s + 2·25-s + 10·29-s + 6·31-s + 8·35-s − 10·37-s + 8·43-s + 3·49-s + 2·53-s − 16·55-s − 4·59-s − 4·61-s − 8·65-s − 12·67-s + 6·71-s − 20·73-s − 8·77-s + 18·79-s − 10·83-s − 8·85-s + 16·89-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s − 1.25·23-s + 2/5·25-s + 1.85·29-s + 1.07·31-s + 1.35·35-s − 1.64·37-s + 1.21·43-s + 3/7·49-s + 0.274·53-s − 2.15·55-s − 0.520·59-s − 0.512·61-s − 0.992·65-s − 1.46·67-s + 0.712·71-s − 2.34·73-s − 0.911·77-s + 2.02·79-s − 1.09·83-s − 0.867·85-s + 1.69·89-s + ⋯

Functional equation

Λ(s)=(91699776s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(91699776s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9169977691699776    =    26⋅34⋅72⋅1922^{6} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 5846.855846.85
Root analytic conductor: 8.744418.74441
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 91699776, ( :1/2,1/2), 1)(4,\ 91699776,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5959263651.595926365
L(12)L(\frac12) ≈\approx 1.5959263651.595926365
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
7C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good5C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C4C_4 1−2T+10T2−2pT3+p2T4 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−2T+18T2−2pT3+p2T4 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+6T+38T2+6pT3+p2T4 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−10T+66T2−10pT3+p2T4 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−6T+54T2−6pT3+p2T4 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+10T+82T2+10pT3+p2T4 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53D4D_{4} 1−2T+90T2−2pT3+p2T4 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67D4D_{4} 1+12T+102T2+12pT3+p2T4 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−6T−2T2−6pT3+p2T4 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 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+10T+174T2+10pT3+p2T4 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−16T+174T2−16pT3+p2T4 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+126T2+p2T4 1 + 126 T^{2} + 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

−7.84951657561159274106672978736, −7.60358830764236590411092016299, −7.07474396074337534150988546293, −6.99337560207161330067812098486, −6.39649675499337916231273539640, −6.31480450206719184619501330049, −5.79645148908963802216614021230, −5.75058348493841252944320472383, −4.84588209817386277044704354395, −4.66293939515861899167726718694, −4.26791466512881524167539492356, −3.97319549997191632117540244165, −3.58789587897799903431276403506, −3.51297941035596890293402275489, −2.82970041661384375884560329137, −2.66144195653829354079566755847, −1.76980991153540768376863713602, −1.52196817292305346432031620228, −0.70153349473876257868841530637, −0.42607731870913130939555465244, 0.42607731870913130939555465244, 0.70153349473876257868841530637, 1.52196817292305346432031620228, 1.76980991153540768376863713602, 2.66144195653829354079566755847, 2.82970041661384375884560329137, 3.51297941035596890293402275489, 3.58789587897799903431276403506, 3.97319549997191632117540244165, 4.26791466512881524167539492356, 4.66293939515861899167726718694, 4.84588209817386277044704354395, 5.75058348493841252944320472383, 5.79645148908963802216614021230, 6.31480450206719184619501330049, 6.39649675499337916231273539640, 6.99337560207161330067812098486, 7.07474396074337534150988546293, 7.60358830764236590411092016299, 7.84951657561159274106672978736

Graph of the ZZ-function along the critical line