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Properties

Label 2-20e2-25.11-c1-0-2
Degree $2$
Conductor $400$
Sign $0.791 - 0.611i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.190 − 0.587i)3-s + (−1.39 + 1.74i)5-s − 1.83·7-s + (2.11 + 1.53i)9-s + (4.19 − 3.04i)11-s + (2.22 + 1.61i)13-s + (0.757 + 1.15i)15-s + (2.02 + 6.23i)17-s + (2.20 + 6.79i)19-s + (−0.350 + 1.07i)21-s + (−3.57 + 2.59i)23-s + (−1.08 − 4.88i)25-s + (2.80 − 2.04i)27-s + (2.24 − 6.89i)29-s + (−0.240 − 0.740i)31-s + ⋯
L(s)  = 1  + (0.110 − 0.339i)3-s + (−0.626 + 0.779i)5-s − 0.692·7-s + (0.706 + 0.512i)9-s + (1.26 − 0.918i)11-s + (0.617 + 0.448i)13-s + (0.195 + 0.298i)15-s + (0.490 + 1.51i)17-s + (0.506 + 1.55i)19-s + (−0.0764 + 0.235i)21-s + (−0.745 + 0.541i)23-s + (−0.216 − 0.976i)25-s + (0.540 − 0.392i)27-s + (0.416 − 1.28i)29-s + (−0.0432 − 0.133i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.791 - 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25225 + 0.427281i\)
\(L(\frac12)\) \(\approx\) \(1.25225 + 0.427281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.39 - 1.74i)T \)
good3 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 1.83T + 7T^{2} \)
11 \( 1 + (-4.19 + 3.04i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.22 - 1.61i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.02 - 6.23i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.20 - 6.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.57 - 2.59i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.24 + 6.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.240 + 0.740i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.60 - 1.89i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.84 + 2.06i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.56T + 43T^{2} \)
47 \( 1 + (1.96 - 6.03i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.74 + 8.44i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.345 + 0.250i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.55 + 1.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.0382 + 0.117i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.97 + 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.472 + 0.343i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.45 - 10.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.47 + 16.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-6.23 + 4.53i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.10 - 6.48i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49664294178602316757665656589, −10.40201199099307545513475086865, −9.745098441552422654394950738976, −8.341657246932230555047168615503, −7.76593345729010661404165365737, −6.46890987167667580068999081868, −6.08939024618332743483920306777, −3.99185855558289963068375562003, −3.49393866008785515063155600057, −1.61256941855808855241610112693, 1.00982803517734375852602730982, 3.20333344415686612582544553280, 4.22237123527859644150776456892, 5.09454672525128985353358227877, 6.68573922035206344954939066692, 7.27088585477268545133103662185, 8.697523311805876774749776238303, 9.370776461308400299065933744719, 9.980918267792404340759480712883, 11.35595629250501435489309046278

Graph of the $Z$-function along the critical line