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Properties

Label 4-186e2-1.1-c1e2-0-4
Degree 44
Conductor 3459634596
Sign 11
Analytic cond. 2.205872.20587
Root an. cond. 1.218691.21869
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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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 + 3·13-s + 4·14-s − 6·15-s + 5·16-s − 17-s + 6·18-s − 19-s + 9·20-s − 4·21-s − 2·22-s − 16·23-s − 8·24-s + 25-s + 6·26-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 + 0.832·13-s + 1.06·14-s − 1.54·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s − 0.229·19-s + 2.01·20-s − 0.872·21-s − 0.426·22-s − 3.33·23-s − 1.63·24-s + 1/5·25-s + 1.17·26-s − 0.769·27-s + 1.13·28-s + ⋯

Functional equation

Λ(s)=(34596s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(34596s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3459634596    =    22⋅32⋅3122^{2} \cdot 3^{2} \cdot 31^{2}
Sign: 11
Analytic conductor: 2.205872.20587
Root analytic conductor: 1.218691.21869
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 34596, ( :1/2,1/2), 1)(4,\ 34596,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6681253742.668125374
L(12)L(\frac12) ≈\approx 2.6681253742.668125374
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}
31C1C_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}
7D4D_{4} 1−2T−2T2−2pT3+p2T4 1 - 2 T - 2 T^{2} - 2 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}
13D4D_{4} 1−3T+24T2−3pT3+p2T4 1 - 3 T + 24 T^{2} - 3 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+T+34T2+pT3+p2T4 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4}
23C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
29D4D_{4} 1+6T+50T2+6pT3+p2T4 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
41D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+10T+94T2+10pT3+p2T4 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−5T+62T2−5pT3+p2T4 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4}
53C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
59D4D_{4} 1−6T+110T2−6pT3+p2T4 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−3T+120T2−3pT3+p2T4 1 - 3 T + 120 T^{2} - 3 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−13T+138T2−13pT3+p2T4 1 - 13 T + 138 T^{2} - 13 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+7T+150T2+7pT3+p2T4 1 + 7 T + 150 T^{2} + 7 p T^{3} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79D4D_{4} 1−5T+126T2−5pT3+p2T4 1 - 5 T + 126 T^{2} - 5 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−5T+66T2−5pT3+p2T4 1 - 5 T + 66 T^{2} - 5 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}
97D4D_{4} 1+9T+108T2+9pT3+p2T4 1 + 9 T + 108 T^{2} + 9 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

−12.82970242883536565766697726296, −12.41632016184871497591587058850, −11.77635719893748808657083014272, −11.64292813372037331744539020433, −10.99724376699698323288834005611, −10.58074293327207340388459008405, −9.960164633792659717568248243671, −9.848792134156071187791953251953, −8.813422363596946949301763839220, −8.020998539187196701231981613739, −7.67343845024421837813346723312, −6.65478381999053354976868805730, −6.33537835797121285546506423012, −5.89527281517259554048250840687, −5.32146829320858463202368391376, −5.08763962406561887319709169070, −4.02831647294226359807140406812, −3.74481057082569996274433617289, −2.02882643892120175759435607569, −1.91959031414303079475692139098, 1.91959031414303079475692139098, 2.02882643892120175759435607569, 3.74481057082569996274433617289, 4.02831647294226359807140406812, 5.08763962406561887319709169070, 5.32146829320858463202368391376, 5.89527281517259554048250840687, 6.33537835797121285546506423012, 6.65478381999053354976868805730, 7.67343845024421837813346723312, 8.020998539187196701231981613739, 8.813422363596946949301763839220, 9.848792134156071187791953251953, 9.960164633792659717568248243671, 10.58074293327207340388459008405, 10.99724376699698323288834005611, 11.64292813372037331744539020433, 11.77635719893748808657083014272, 12.41632016184871497591587058850, 12.82970242883536565766697726296

Graph of the ZZ-function along the critical line