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Properties

Label 2-20e2-25.14-c1-0-12
Degree $2$
Conductor $400$
Sign $0.928 + 0.370i$
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  + (2.92 + 0.949i)3-s + (−0.820 − 2.08i)5-s − 4.49i·7-s + (5.20 + 3.78i)9-s + (1.31 − 0.953i)11-s + (−2.58 + 3.55i)13-s + (−0.421 − 6.85i)15-s + (−0.478 + 0.155i)17-s + (−0.574 − 1.76i)19-s + (4.26 − 13.1i)21-s + (1.96 + 2.71i)23-s + (−3.65 + 3.41i)25-s + (6.19 + 8.52i)27-s + (−1.88 + 5.78i)29-s + (1.80 + 5.56i)31-s + ⋯
L(s)  = 1  + (1.68 + 0.547i)3-s + (−0.366 − 0.930i)5-s − 1.69i·7-s + (1.73 + 1.26i)9-s + (0.395 − 0.287i)11-s + (−0.717 + 0.987i)13-s + (−0.108 − 1.76i)15-s + (−0.115 + 0.0376i)17-s + (−0.131 − 0.405i)19-s + (0.931 − 2.86i)21-s + (0.410 + 0.565i)23-s + (−0.730 + 0.682i)25-s + (1.19 + 1.64i)27-s + (−0.349 + 1.07i)29-s + (0.324 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.928 + 0.370i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.928 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14596 - 0.412450i\)
\(L(\frac12)\) \(\approx\) \(2.14596 - 0.412450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.820 + 2.08i)T \)
good3 \( 1 + (-2.92 - 0.949i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + 4.49iT - 7T^{2} \)
11 \( 1 + (-1.31 + 0.953i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.58 - 3.55i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.478 - 0.155i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.574 + 1.76i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.96 - 2.71i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.88 - 5.78i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.80 - 5.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.30 - 3.17i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.52 - 1.83i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.10iT - 43T^{2} \)
47 \( 1 + (-4.98 - 1.62i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.22 - 1.37i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.70 - 2.69i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.70 - 1.24i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.05 - 1.31i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.99 + 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.78 + 12.0i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.66 + 8.21i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.83 - 3.19i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.48 + 1.80i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.30 + 0.749i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.99476141472824130850257451922, −10.07196299188942195199970867995, −9.205474308327866241668606466955, −8.668568769947867335432629711106, −7.57669822540361773272386639081, −7.02566793554936064335100803914, −4.78392889320898025198772290500, −4.16466281409025568264500780774, −3.27719765787450228542452884455, −1.49958445627628425249918382078, 2.31555370795371207918466207381, 2.72256632264231374286979001727, 3.97250908873837946156510782388, 5.74366217093173441868190491326, 6.89286364194430209746365134386, 7.80649754351225278766292298404, 8.453255540157008329261551478704, 9.350173017032471817792647928564, 10.10775454123513760321409408430, 11.54097609670805955639456090359

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