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Properties

Label 4-48e4-1.1-c1e2-0-33
Degree 44
Conductor 53084165308416
Sign 11
Analytic cond. 338.469338.469
Root an. cond. 4.289234.28923
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·17-s − 8·19-s + 2·25-s − 12·41-s − 8·43-s − 2·49-s − 24·59-s + 8·67-s − 4·73-s + 12·89-s − 4·97-s − 24·107-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.91·17-s − 1.83·19-s + 2/5·25-s − 1.87·41-s − 1.21·43-s − 2/7·49-s − 3.12·59-s + 0.977·67-s − 0.468·73-s + 1.27·89-s − 0.406·97-s − 2.32·107-s + 1.12·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(5308416s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5308416s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 53084165308416    =    216⋅342^{16} \cdot 3^{4}
Sign: 11
Analytic conductor: 338.469338.469
Root analytic conductor: 4.289234.28923
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 5308416, ( :1/2,1/2), 1)(4,\ 5308416,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
53C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
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

−9.015531523845900068320152927652, −8.443354404956822832722380798542, −8.057050852209817655105485886173, −7.86852777929663488987876925219, −7.08867804479557123672170829895, −6.68571996650659968785279685501, −6.58327905646539781450949091397, −6.32027506245078211169784598159, −5.73825499223907028216052868382, −5.06182597172327518291954383062, −4.63929477064860348917448634603, −4.63027211397494488448464931756, −3.87610903325282435480477851545, −3.63535021906343364664894789563, −2.77122505365859789286595438287, −2.49260455118983694588862785334, −1.85410503708777745297468497660, −1.49968114299870965218229307264, 0, 0, 1.49968114299870965218229307264, 1.85410503708777745297468497660, 2.49260455118983694588862785334, 2.77122505365859789286595438287, 3.63535021906343364664894789563, 3.87610903325282435480477851545, 4.63027211397494488448464931756, 4.63929477064860348917448634603, 5.06182597172327518291954383062, 5.73825499223907028216052868382, 6.32027506245078211169784598159, 6.58327905646539781450949091397, 6.68571996650659968785279685501, 7.08867804479557123672170829895, 7.86852777929663488987876925219, 8.057050852209817655105485886173, 8.443354404956822832722380798542, 9.015531523845900068320152927652

Graph of the ZZ-function along the critical line