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Properties

Label 4-2352e2-1.1-c1e2-0-45
Degree 44
Conductor 55319045531904
Sign 11
Analytic cond. 352.718352.718
Root an. cond. 4.333684.33368
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 + 4·5-s + 2·11-s + 12·13-s + 4·15-s − 4·17-s + 4·19-s + 2·23-s + 5·25-s − 27-s − 4·29-s + 2·33-s − 2·37-s + 12·39-s + 8·43-s − 12·47-s − 4·51-s + 6·53-s + 8·55-s + 4·57-s + 8·59-s + 6·61-s + 48·65-s − 8·67-s + 2·69-s − 28·71-s − 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s + 0.603·11-s + 3.32·13-s + 1.03·15-s − 0.970·17-s + 0.917·19-s + 0.417·23-s + 25-s − 0.192·27-s − 0.742·29-s + 0.348·33-s − 0.328·37-s + 1.92·39-s + 1.21·43-s − 1.75·47-s − 0.560·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.04·59-s + 0.768·61-s + 5.95·65-s − 0.977·67-s + 0.240·69-s − 3.32·71-s − 0.234·73-s + ⋯

Functional equation

Λ(s)=(5531904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5531904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 55319045531904    =    28⋅32⋅742^{8} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 352.718352.718
Root analytic conductor: 4.333684.33368
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5531904, ( :1/2,1/2), 1)(4,\ 5531904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.4846785826.484678582
L(12)L(\frac12) ≈\approx 6.4846785826.484678582
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
3C2C_2 1−T+T2 1 - T + T^{2}
7 1 1
good5C22C_2^2 1−4T+11T2−4pT3+p2T4 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−2T−7T2−2pT3+p2T4 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
17C22C_2^2 1+4T−T2+4pT3+p2T4 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−4T−3T2−4pT3+p2T4 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−2T−19T2−2pT3+p2T4 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
37C22C_2^2 1+2T−33T2+2pT3+p2T4 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1+12T+97T2+12pT3+p2T4 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−8T+5T2−8pT3+p2T4 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−6T−25T2−6pT3+p2T4 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+8T−3T2+8pT3+p2T4 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4}
71C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
73C22C_2^2 1+2T−69T2+2pT3+p2T4 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−12T+65T2−12pT3+p2T4 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4}
83C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
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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

−9.027555529438363187943283828927, −8.956046840235688403531788546853, −8.482407516619999501636047128059, −8.376111132625193368952849482231, −7.46728862802796905732601872059, −7.39982098739445310535291862525, −6.66882391126197595563885479017, −6.34511148292552063414078427128, −6.08400937354671464385571117165, −5.66918592988547226079585033228, −5.59165735081104321113295012941, −4.75917910687312562857920022937, −4.33448063768900240479575768496, −3.76178850314619155644988262670, −3.38640234730325422353778433222, −3.09217022834732506857302488714, −2.24017521120147243938185483902, −1.87530192871213229516472357084, −1.39989156679247641154991923106, −0.913352928619763097449624431112, 0.913352928619763097449624431112, 1.39989156679247641154991923106, 1.87530192871213229516472357084, 2.24017521120147243938185483902, 3.09217022834732506857302488714, 3.38640234730325422353778433222, 3.76178850314619155644988262670, 4.33448063768900240479575768496, 4.75917910687312562857920022937, 5.59165735081104321113295012941, 5.66918592988547226079585033228, 6.08400937354671464385571117165, 6.34511148292552063414078427128, 6.66882391126197595563885479017, 7.39982098739445310535291862525, 7.46728862802796905732601872059, 8.376111132625193368952849482231, 8.482407516619999501636047128059, 8.956046840235688403531788546853, 9.027555529438363187943283828927

Graph of the ZZ-function along the critical line