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Properties

Label 2-20e2-25.6-c1-0-13
Degree $2$
Conductor $400$
Sign $-0.186 + 0.982i$
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  + (1.30 − 0.951i)3-s + (0.279 − 2.21i)5-s − 3.77·7-s + (−0.118 + 0.363i)9-s + (−1.25 − 3.86i)11-s + (2.07 − 6.39i)13-s + (−1.74 − 3.16i)15-s + (5.08 + 3.69i)17-s + (−0.588 − 0.427i)19-s + (−4.94 + 3.58i)21-s + (−0.363 − 1.11i)23-s + (−4.84 − 1.23i)25-s + (1.69 + 5.20i)27-s + (−2.91 + 2.11i)29-s + (4.91 + 3.56i)31-s + ⋯
L(s)  = 1  + (0.755 − 0.549i)3-s + (0.124 − 0.992i)5-s − 1.42·7-s + (−0.0393 + 0.121i)9-s + (−0.378 − 1.16i)11-s + (0.576 − 1.77i)13-s + (−0.450 − 0.818i)15-s + (1.23 + 0.895i)17-s + (−0.134 − 0.0980i)19-s + (−1.07 + 0.783i)21-s + (−0.0757 − 0.233i)23-s + (−0.968 − 0.247i)25-s + (0.325 + 1.00i)27-s + (−0.540 + 0.392i)29-s + (0.882 + 0.640i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.186 + 0.982i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.186 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921273 - 1.11249i\)
\(L(\frac12)\) \(\approx\) \(0.921273 - 1.11249i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.279 + 2.21i)T \)
good3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 3.77T + 7T^{2} \)
11 \( 1 + (1.25 + 3.86i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.07 + 6.39i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-5.08 - 3.69i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.588 + 0.427i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.363 + 1.11i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.91 - 2.11i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.91 - 3.56i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.77 + 5.45i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.460 - 1.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (-2.65 + 1.93i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.31 - 0.958i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.207 + 0.639i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.26 - 6.95i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.65 + 3.38i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.00 - 1.45i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.14 - 6.61i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.24 + 1.63i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.14 - 3.01i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.743 + 2.28i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.45 + 2.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.81223493054353088341087712950, −10.08615717030385414433396675522, −8.939726689632421421982952147180, −8.296566726106524390400015054062, −7.60267253300454735664699866021, −6.03011926829415138433413923873, −5.51027676288472315476252160832, −3.61572965280909319962375233288, −2.82177285970517874740158824208, −0.893479526366073641665584119020, 2.42017693288289693731967796336, 3.38173138510002729920379863735, 4.28364924427226066364477297191, 6.05809930932967411648882010661, 6.80882308856709684116929425444, 7.71994681674431745585555701082, 9.270602987088500015520933595582, 9.595763134067997474933843170610, 10.25392836391134818831290789036, 11.54854614137377412954980841856

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