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Properties

Label 4-352e2-1.1-c1e2-0-3
Degree 44
Conductor 123904123904
Sign 11
Analytic cond. 7.900227.90022
Root an. cond. 1.676521.67652
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  − 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)=(123904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 123904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(123904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 123904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 123904123904    =    210⋅1122^{10} \cdot 11^{2}
Sign: 11
Analytic conductor: 7.900227.90022
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 123904, ( :1/2,1/2), 1)(4,\ 123904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6731568471.673156847
L(12)L(\frac12) ≈\approx 1.6731568471.673156847
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

−11.71637930315705832383908896038, −11.32072987439750756990133324195, −10.55797848962274206480390322282, −10.31974112410654281292072961653, −10.06589824870708370937970806103, −9.600970605761722555739124403564, −8.720317118500883302013183754184, −8.639658734738239079781631052209, −8.017846064032105851810084781806, −7.52032751894912668633406699865, −6.51405201310665444503786570704, −6.42574959118232184729111613601, −5.85574000121122320487660419198, −5.57052142861751043992859836001, −4.80128707043152207413273055438, −4.39438615218903394977724440261, −3.30198380786633657028182233539, −2.81883750856868059259196740156, −1.93278793289298093103504980717, −1.00310502704205625408320475655, 1.00310502704205625408320475655, 1.93278793289298093103504980717, 2.81883750856868059259196740156, 3.30198380786633657028182233539, 4.39438615218903394977724440261, 4.80128707043152207413273055438, 5.57052142861751043992859836001, 5.85574000121122320487660419198, 6.42574959118232184729111613601, 6.51405201310665444503786570704, 7.52032751894912668633406699865, 8.017846064032105851810084781806, 8.639658734738239079781631052209, 8.720317118500883302013183754184, 9.600970605761722555739124403564, 10.06589824870708370937970806103, 10.31974112410654281292072961653, 10.55797848962274206480390322282, 11.32072987439750756990133324195, 11.71637930315705832383908896038

Graph of the ZZ-function along the critical line