Properties

Label 2-20e2-1.1-c3-0-4
Degree 22
Conductor 400400
Sign 11
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 26·7-s − 26·9-s − 45·11-s + 44·13-s + 117·17-s + 91·19-s + 26·21-s + 18·23-s + 53·27-s + 144·29-s − 26·31-s + 45·33-s − 214·37-s − 44·39-s − 459·41-s + 460·43-s + 468·47-s + 333·49-s − 117·51-s + 558·53-s − 91·57-s + 72·59-s − 118·61-s + 676·63-s − 251·67-s − 18·69-s + ⋯
L(s)  = 1  − 0.192·3-s − 1.40·7-s − 0.962·9-s − 1.23·11-s + 0.938·13-s + 1.66·17-s + 1.09·19-s + 0.270·21-s + 0.163·23-s + 0.377·27-s + 0.922·29-s − 0.150·31-s + 0.237·33-s − 0.950·37-s − 0.180·39-s − 1.74·41-s + 1.63·43-s + 1.45·47-s + 0.970·49-s − 0.321·51-s + 1.44·53-s − 0.211·57-s + 0.158·59-s − 0.247·61-s + 1.35·63-s − 0.457·67-s − 0.0314·69-s + ⋯

Functional equation

Λ(s)=(400s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(400s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24522^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 400, ( :3/2), 1)(2,\ 400,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) \approx 1.2054528611.205452861
L(12)L(\frac12) \approx 1.2054528611.205452861
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+T+p3T2 1 + T + p^{3} T^{2}
7 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
11 1+45T+p3T2 1 + 45 T + p^{3} T^{2}
13 144T+p3T2 1 - 44 T + p^{3} T^{2}
17 1117T+p3T2 1 - 117 T + p^{3} T^{2}
19 191T+p3T2 1 - 91 T + p^{3} T^{2}
23 118T+p3T2 1 - 18 T + p^{3} T^{2}
29 1144T+p3T2 1 - 144 T + p^{3} T^{2}
31 1+26T+p3T2 1 + 26 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1+459T+p3T2 1 + 459 T + p^{3} T^{2}
43 1460T+p3T2 1 - 460 T + p^{3} T^{2}
47 1468T+p3T2 1 - 468 T + p^{3} T^{2}
53 1558T+p3T2 1 - 558 T + p^{3} T^{2}
59 172T+p3T2 1 - 72 T + p^{3} T^{2}
61 1+118T+p3T2 1 + 118 T + p^{3} T^{2}
67 1+251T+p3T2 1 + 251 T + p^{3} T^{2}
71 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
73 1299T+p3T2 1 - 299 T + p^{3} T^{2}
79 1898T+p3T2 1 - 898 T + p^{3} T^{2}
83 1+927T+p3T2 1 + 927 T + p^{3} T^{2}
89 1351T+p3T2 1 - 351 T + p^{3} T^{2}
97 1386T+p3T2 1 - 386 T + p^{3} T^{2}
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   L(s)=p j=12(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.66222305581732317797491050676, −10.08665060484949798492308686817, −9.058462953622159937205941410219, −8.100774325401966563103217330172, −7.06472333355930260135293296809, −5.87965745953089203269089337582, −5.36091428648694381763533521097, −3.50693435780534881528933857626, −2.84020271325597463650178054580, −0.70634813288964357001921596267, 0.70634813288964357001921596267, 2.84020271325597463650178054580, 3.50693435780534881528933857626, 5.36091428648694381763533521097, 5.87965745953089203269089337582, 7.06472333355930260135293296809, 8.100774325401966563103217330172, 9.058462953622159937205941410219, 10.08665060484949798492308686817, 10.66222305581732317797491050676

Graph of the ZZ-function along the critical line