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Properties

Label 2-20e2-100.27-c1-0-6
Degree $2$
Conductor $400$
Sign $0.425 + 0.904i$
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  + (−2.58 + 1.31i)3-s − 2.23·5-s + (−1.62 + 1.62i)7-s + (3.18 − 4.37i)9-s + (1.12 + 1.54i)11-s + (−0.951 + 0.150i)13-s + (5.77 − 2.94i)15-s + (2.48 − 4.88i)17-s + (−1.87 − 5.77i)19-s + (2.06 − 6.34i)21-s + (−2.55 − 0.404i)23-s + 5.00·25-s + (−1.09 + 6.90i)27-s + (2.65 + 0.861i)29-s + (9.86 − 3.20i)31-s + ⋯
L(s)  = 1  + (−1.49 + 0.760i)3-s − 0.999·5-s + (−0.615 + 0.615i)7-s + (1.06 − 1.45i)9-s + (0.339 + 0.466i)11-s + (−0.263 + 0.0417i)13-s + (1.49 − 0.760i)15-s + (0.603 − 1.18i)17-s + (−0.430 − 1.32i)19-s + (0.450 − 1.38i)21-s + (−0.532 − 0.0843i)23-s + 1.00·25-s + (−0.210 + 1.32i)27-s + (0.492 + 0.159i)29-s + (1.77 − 0.575i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.279617 - 0.177450i\)
\(L(\frac12)\) \(\approx\) \(0.279617 - 0.177450i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
good3 \( 1 + (2.58 - 1.31i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (1.62 - 1.62i)T - 7iT^{2} \)
11 \( 1 + (-1.12 - 1.54i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.951 - 0.150i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-2.48 + 4.88i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.87 + 5.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.55 + 0.404i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-2.65 - 0.861i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.86 + 3.20i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.58 - 10.0i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (8.52 + 6.19i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.17 + 6.17i)T + 43iT^{2} \)
47 \( 1 + (0.494 + 0.970i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (4.26 + 8.36i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (7.85 + 5.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.975 - 0.708i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-5.77 - 2.94i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (8.83 + 2.86i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.212 - 1.33i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.28 - 7.03i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.97 + 13.6i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (1.95 + 2.69i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.23 + 1.14i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.34415341487433748711279601501, −10.21446666169037340692947248488, −9.611747287083550504178641731383, −8.448470735353908342746635234990, −7.01442924245737906849313694813, −6.37176211470175072342438660070, −5.04683816051515521081768177726, −4.51048541631331103852482156445, −3.09938558785023656372961242163, −0.30645057432516719204356960589, 1.19369385646964825319863719351, 3.53857502896142536213495793807, 4.62701863020925596358456011152, 6.02831361188981854030731189070, 6.51391185002197257276099705120, 7.61249461611494749176006360816, 8.311333897930613448455425148087, 10.08180004867959843851584984416, 10.63802600379768044282549607043, 11.61291262486260752309685462949

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