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Properties

Label 4-546e2-1.1-c1e2-0-66
Degree 44
Conductor 298116298116
Sign 11
Analytic cond. 19.008119.0081
Root an. cond. 2.088022.08802
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  + 2·2-s + 2·3-s + 3·4-s + 3·5-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s + 6·10-s + 11-s + 6·12-s + 2·13-s − 4·14-s + 6·15-s + 5·16-s − 17-s + 6·18-s + 3·19-s + 9·20-s − 4·21-s + 2·22-s − 7·23-s + 8·24-s + 25-s + 4·26-s + 4·27-s − 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.34·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 1.89·10-s + 0.301·11-s + 1.73·12-s + 0.554·13-s − 1.06·14-s + 1.54·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s + 0.688·19-s + 2.01·20-s − 0.872·21-s + 0.426·22-s − 1.45·23-s + 1.63·24-s + 1/5·25-s + 0.784·26-s + 0.769·27-s − 1.13·28-s + ⋯

Functional equation

Λ(s)=(298116s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(298116s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 298116298116    =    22⋅32⋅72⋅1322^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 19.008119.0081
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 298116, ( :1/2,1/2), 1)(4,\ 298116,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 8.0665347398.066534739
L(12)L(\frac12) ≈\approx 8.0665347398.066534739
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}
3C1C_1 (1−T)2 ( 1 - T )^{2}
7C1C_1 (1+T)2 ( 1 + T )^{2}
13C1C_1 (1−T)2 ( 1 - T )^{2}
good5C22C_2^2 1−3T+8T2−3pT3+p2T4 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4}
11D4D_{4} 1−T+18T2−pT3+p2T4 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4}
17D4D_{4} 1+T−4T2+pT3+p2T4 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4}
19D4D_{4} 1−3T+2T2−3pT3+p2T4 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+7T+54T2+7pT3+p2T4 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−T+20T2−pT3+p2T4 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4}
31D4D_{4} 1+4T−2T2+4pT3+p2T4 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+11T+100T2+11pT3+p2T4 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−6T+74T2−6pT3+p2T4 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+17T+154T2+17pT3+p2T4 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53D4D_{4} 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+12T+86T2+12pT3+p2T4 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−T+84T2−pT3+p2T4 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4}
67D4D_{4} 1+6T+126T2+6pT3+p2T4 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73D4D_{4} 1+3T+144T2+3pT3+p2T4 1 + 3 T + 144 T^{2} + 3 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+14T+190T2+14pT3+p2T4 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−26T+318T2−26pT3+p2T4 1 - 26 T + 318 T^{2} - 26 p T^{3} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97D4D_{4} 1+4T−74T2+4pT3+p2T4 1 + 4 T - 74 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

−10.96102185939495039898716611437, −10.41857078250021618048528429289, −10.22060041031839152001460924305, −9.756888510714417580519514037867, −9.233370560023010947145845166715, −9.010485694780640031278337377431, −8.305995732921860230626379067102, −7.83825761795388458926986676872, −7.32618673380141590440426968804, −6.71997963436205347221010032644, −6.32928801133355306306372815819, −6.09386238561401144317148233861, −5.20605424187117081970844569536, −5.20040415974629017633015291518, −4.13517339401693938788755970479, −3.73522433295182470573736754474, −3.32218683883157689035440528794, −2.67685834892058680739712973755, −1.96532787879004213571976830321, −1.61386041996840783631232130153, 1.61386041996840783631232130153, 1.96532787879004213571976830321, 2.67685834892058680739712973755, 3.32218683883157689035440528794, 3.73522433295182470573736754474, 4.13517339401693938788755970479, 5.20040415974629017633015291518, 5.20605424187117081970844569536, 6.09386238561401144317148233861, 6.32928801133355306306372815819, 6.71997963436205347221010032644, 7.32618673380141590440426968804, 7.83825761795388458926986676872, 8.305995732921860230626379067102, 9.010485694780640031278337377431, 9.233370560023010947145845166715, 9.756888510714417580519514037867, 10.22060041031839152001460924305, 10.41857078250021618048528429289, 10.96102185939495039898716611437

Graph of the ZZ-function along the critical line