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Properties

Label 4-3240e2-1.1-c1e2-0-2
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
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  − 5-s − 4·7-s + 6·13-s + 4·17-s + 8·19-s − 8·23-s − 6·29-s + 4·35-s − 12·37-s + 10·41-s + 4·43-s + 8·47-s + 7·49-s − 20·53-s − 6·61-s − 6·65-s + 4·67-s − 28·73-s − 16·79-s + 12·83-s − 4·85-s − 4·89-s − 24·91-s − 8·95-s − 2·97-s − 14·101-s − 4·103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 1.66·13-s + 0.970·17-s + 1.83·19-s − 1.66·23-s − 1.11·29-s + 0.676·35-s − 1.97·37-s + 1.56·41-s + 0.609·43-s + 1.16·47-s + 49-s − 2.74·53-s − 0.768·61-s − 0.744·65-s + 0.488·67-s − 3.27·73-s − 1.80·79-s + 1.31·83-s − 0.433·85-s − 0.423·89-s − 2.51·91-s − 0.820·95-s − 0.203·97-s − 1.39·101-s − 0.394·103-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.86903237020.8690323702
L(12)L(\frac12) ≈\approx 0.86903237020.8690323702
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
5C2C_2 1+T+T2 1 + T + T^{2}
good7C2C_2 (1−T+pT2)(1+5T+pT2) ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} )
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1+8T+41T2+8pT3+p2T4 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+6T+7T2+6pT3+p2T4 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C22C_2^2 1−10T+59T2−10pT3+p2T4 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−8T+17T2−8pT3+p2T4 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + 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 1−4T−51T2−4pT3+p2T4 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
79C22C_2^2 1+16T+177T2+16pT3+p2T4 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−12T+61T2−12pT3+p2T4 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97C22C_2^2 1+2T−93T2+2pT3+p2T4 1 + 2 T - 93 T^{2} + 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

−8.879165441675549819162479000339, −8.486890620499976171795781357785, −8.019354862040483284786881520762, −7.66709261725692101914901298622, −7.43256133003818657128492178788, −7.03743928303854781838466860022, −6.56385533080558562704276352189, −5.99508919014626089323380956792, −5.97533616059358178362659850069, −5.55916685728081394932689784937, −5.22156391908365394800213222384, −4.25774743919011393031473327542, −4.22526218927235616495563985378, −3.52020820957182122929711149374, −3.40951499354888937676908472563, −3.03827374883899310747033426176, −2.46445452097762539289244494149, −1.40537062787603658694890605059, −1.40383064709506987211565168120, −0.30027630621290779772511235307, 0.30027630621290779772511235307, 1.40383064709506987211565168120, 1.40537062787603658694890605059, 2.46445452097762539289244494149, 3.03827374883899310747033426176, 3.40951499354888937676908472563, 3.52020820957182122929711149374, 4.22526218927235616495563985378, 4.25774743919011393031473327542, 5.22156391908365394800213222384, 5.55916685728081394932689784937, 5.97533616059358178362659850069, 5.99508919014626089323380956792, 6.56385533080558562704276352189, 7.03743928303854781838466860022, 7.43256133003818657128492178788, 7.66709261725692101914901298622, 8.019354862040483284786881520762, 8.486890620499976171795781357785, 8.879165441675549819162479000339

Graph of the ZZ-function along the critical line