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Properties

Label 2-20e2-1.1-c3-0-1
Degree 22
Conductor 400400
Sign 11
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.71·3-s − 8.71·7-s + 49.0·9-s − 20·11-s − 52.3·13-s − 69.7·17-s − 84·19-s + 76.0·21-s − 61.0·23-s − 191.·27-s − 6·29-s + 224·31-s + 174.·33-s − 122.·37-s + 456.·39-s + 266·41-s + 305.·43-s + 374.·47-s − 267·49-s + 608.·51-s − 366.·53-s + 732.·57-s − 28·59-s + 182·61-s − 427.·63-s + 427.·67-s + 532.·69-s + ⋯
L(s)  = 1  − 1.67·3-s − 0.470·7-s + 1.81·9-s − 0.548·11-s − 1.11·13-s − 0.995·17-s − 1.01·19-s + 0.789·21-s − 0.553·23-s − 1.36·27-s − 0.0384·29-s + 1.29·31-s + 0.919·33-s − 0.542·37-s + 1.87·39-s + 1.01·41-s + 1.08·43-s + 1.16·47-s − 0.778·49-s + 1.66·51-s − 0.948·53-s + 1.70·57-s − 0.0617·59-s + 0.382·61-s − 0.854·63-s + 0.778·67-s + 0.928·69-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\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    =    24⋅522^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: no
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 0.48025717300.4802571730
L(12)L(\frac12) ≈\approx 0.48025717300.4802571730
L(52)L(\frac{5}{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}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+8.71T+27T2 1 + 8.71T + 27T^{2}
7 1+8.71T+343T2 1 + 8.71T + 343T^{2}
11 1+20T+1.33e3T2 1 + 20T + 1.33e3T^{2}
13 1+52.3T+2.19e3T2 1 + 52.3T + 2.19e3T^{2}
17 1+69.7T+4.91e3T2 1 + 69.7T + 4.91e3T^{2}
19 1+84T+6.85e3T2 1 + 84T + 6.85e3T^{2}
23 1+61.0T+1.21e4T2 1 + 61.0T + 1.21e4T^{2}
29 1+6T+2.43e4T2 1 + 6T + 2.43e4T^{2}
31 1−224T+2.97e4T2 1 - 224T + 2.97e4T^{2}
37 1+122.T+5.06e4T2 1 + 122.T + 5.06e4T^{2}
41 1−266T+6.89e4T2 1 - 266T + 6.89e4T^{2}
43 1−305.T+7.95e4T2 1 - 305.T + 7.95e4T^{2}
47 1−374.T+1.03e5T2 1 - 374.T + 1.03e5T^{2}
53 1+366.T+1.48e5T2 1 + 366.T + 1.48e5T^{2}
59 1+28T+2.05e5T2 1 + 28T + 2.05e5T^{2}
61 1−182T+2.26e5T2 1 - 182T + 2.26e5T^{2}
67 1−427.T+3.00e5T2 1 - 427.T + 3.00e5T^{2}
71 1+408T+3.57e5T2 1 + 408T + 3.57e5T^{2}
73 1−1.08e3T+3.89e5T2 1 - 1.08e3T + 3.89e5T^{2}
79 1−48T+4.93e5T2 1 - 48T + 4.93e5T^{2}
83 1−200.T+5.71e5T2 1 - 200.T + 5.71e5T^{2}
89 1−1.52e3T+7.04e5T2 1 - 1.52e3T + 7.04e5T^{2}
97 1+557.T+9.12e5T2 1 + 557.T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−10.81452843163982808038560960039, −10.26671805452342738592084242983, −9.282833413302734555711824787553, −7.85400388696147845491978591919, −6.76308618910737609690013885051, −6.13055827566084910750344003704, −5.06917790250011866328137380557, −4.26485984579328047422001422113, −2.35426515648302918600159376835, −0.46738346301242291680060142846, 0.46738346301242291680060142846, 2.35426515648302918600159376835, 4.26485984579328047422001422113, 5.06917790250011866328137380557, 6.13055827566084910750344003704, 6.76308618910737609690013885051, 7.85400388696147845491978591919, 9.282833413302734555711824787553, 10.26671805452342738592084242983, 10.81452843163982808038560960039

Graph of the ZZ-function along the critical line