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Properties

Label 4-1620e2-1.1-c1e2-0-15
Degree 44
Conductor 26244002624400
Sign 11
Analytic cond. 167.334167.334
Root an. cond. 3.596633.59663
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s + 4·13-s − 6·17-s − 2·19-s + 3·25-s − 12·29-s − 2·31-s + 4·35-s − 2·37-s − 12·41-s + 10·43-s − 6·47-s − 8·49-s − 18·53-s − 12·59-s − 8·61-s − 8·65-s − 8·67-s − 12·71-s + 10·73-s − 8·79-s − 6·83-s + 12·85-s − 8·91-s + 4·95-s − 2·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 3/5·25-s − 2.22·29-s − 0.359·31-s + 0.676·35-s − 0.328·37-s − 1.87·41-s + 1.52·43-s − 0.875·47-s − 8/7·49-s − 2.47·53-s − 1.56·59-s − 1.02·61-s − 0.992·65-s − 0.977·67-s − 1.42·71-s + 1.17·73-s − 0.900·79-s − 0.658·83-s + 1.30·85-s − 0.838·91-s + 0.410·95-s − 0.203·97-s + ⋯

Functional equation

Λ(s)=(2624400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2624400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 26244002624400    =    24⋅38⋅522^{4} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 167.334167.334
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2624400, ( :1/2,1/2), 1)(4,\ 2624400,\ (\ :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
5C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1+2T+12T2+2pT3+p2T4 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
13D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+6T+40T2+6pT3+p2T4 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+2T+27T2+2pT3+p2T4 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
29D4D_{4} 1+12T+91T2+12pT3+p2T4 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+2T+15T2+2pT3+p2T4 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+2T+48T2+2pT3+p2T4 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+12T+91T2+12pT3+p2T4 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−10T+108T2−10pT3+p2T4 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+100T2+6pT3+p2T4 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+18T+184T2+18pT3+p2T4 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+12T+151T2+12pT3+p2T4 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67D4D_{4} 1+8T+42T2+8pT3+p2T4 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+12T+151T2+12pT3+p2T4 1 + 12 T + 151 T^{2} + 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−10T+144T2−10pT3+p2T4 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+66T2+8pT3+p2T4 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+6T+28T2+6pT3+p2T4 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+151T2+p2T4 1 + 151 T^{2} + p^{2} T^{4}
97D4D_{4} 1+2T+192T2+2pT3+p2T4 1 + 2 T + 192 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

−9.110609864033189596256480630968, −8.798840750292056172938839657655, −8.523972424114836906283845010353, −7.900682435361490441100623140208, −7.69392605497610704803702373014, −7.23672528186723219200210890434, −6.62949699463280974395363368609, −6.51536639185737443296617170496, −6.01392049417018046996752699513, −5.64426769097997952436996383696, −4.84235530744126004225656247974, −4.67016232255040115722913867398, −3.92384276627848462930723316495, −3.80864020236994072163074395159, −3.09473851804565389652958363628, −2.90305631375347707399054116567, −1.71908120656316357702035224078, −1.63733370336954457494169648636, 0, 0, 1.63733370336954457494169648636, 1.71908120656316357702035224078, 2.90305631375347707399054116567, 3.09473851804565389652958363628, 3.80864020236994072163074395159, 3.92384276627848462930723316495, 4.67016232255040115722913867398, 4.84235530744126004225656247974, 5.64426769097997952436996383696, 6.01392049417018046996752699513, 6.51536639185737443296617170496, 6.62949699463280974395363368609, 7.23672528186723219200210890434, 7.69392605497610704803702373014, 7.900682435361490441100623140208, 8.523972424114836906283845010353, 8.798840750292056172938839657655, 9.110609864033189596256480630968

Graph of the ZZ-function along the critical line