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Properties

Label 2-20e2-25.11-c1-0-7
Degree $2$
Conductor $400$
Sign $0.918 + 0.395i$
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 + (2.09 − 0.792i)5-s + 0.833·7-s + (2.11 + 1.53i)9-s + (−1.45 + 1.05i)11-s + (0.892 + 0.648i)13-s + (−0.0665 − 1.38i)15-s + (−0.642 − 1.97i)17-s + (−1.28 − 3.94i)19-s + (0.159 − 0.489i)21-s + (2.07 − 1.50i)23-s + (3.74 − 3.31i)25-s + (2.80 − 2.04i)27-s + (−0.740 + 2.27i)29-s + (2.74 + 8.43i)31-s + ⋯
L(s)  = 1  + (0.110 − 0.339i)3-s + (0.935 − 0.354i)5-s + 0.314·7-s + (0.706 + 0.512i)9-s + (−0.439 + 0.318i)11-s + (0.247 + 0.179i)13-s + (−0.0171 − 0.356i)15-s + (−0.155 − 0.479i)17-s + (−0.294 − 0.905i)19-s + (0.0347 − 0.106i)21-s + (0.432 − 0.314i)23-s + (0.748 − 0.662i)25-s + (0.540 − 0.392i)27-s + (−0.137 + 0.423i)29-s + (0.492 + 1.51i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.918 + 0.395i$
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: \((2,\ 400,\ (\ :1/2),\ 0.918 + 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69998 - 0.350197i\)
\(L(\frac12)\) \(\approx\) \(1.69998 - 0.350197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 0.792i)T \)
good3 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.833T + 7T^{2} \)
11 \( 1 + (1.45 - 1.05i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.892 - 0.648i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.642 + 1.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.28 + 3.94i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.07 + 1.50i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.740 - 2.27i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.74 - 8.43i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.69 + 6.31i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.51 + 1.09i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (0.628 - 1.93i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.08 - 6.40i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.8 + 7.85i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.601 - 0.436i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.37 + 4.22i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.32 - 10.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.01 - 2.19i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.96 - 12.2i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.49 + 7.67i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (8.54 - 6.21i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.03 + 9.33i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.99334228717549303301051319512, −10.39480007143094311343835809233, −9.315928990584844072548130517266, −8.573485943852512873597177170572, −7.37848318635058730289680169264, −6.61101058880754029560661670073, −5.27488056641540394721382570256, −4.55630181862375729570800331442, −2.65810376326714621252075213777, −1.49078946181636869187082716643, 1.65988975625989442354198002752, 3.16068793733412155025153961495, 4.39769691382279925720180751726, 5.66401610545119181790508260879, 6.45378844503605673519028144624, 7.63759558879178464304443687378, 8.699867843302238900022396909574, 9.687766122096483074269798934356, 10.31747756079943206971128717202, 11.10237126660072629223596013338

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