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Properties

Label 2-20e2-80.69-c1-0-32
Degree $2$
Conductor $400$
Sign $-0.895 - 0.445i$
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  + (0.710 − 1.22i)2-s + (−1.09 − 1.09i)3-s + (−0.991 − 1.73i)4-s + (−2.11 + 0.560i)6-s − 0.973·7-s + (−2.82 − 0.0202i)8-s − 0.616i·9-s + (1.40 + 1.40i)11-s + (−0.813 + 2.97i)12-s + (−4.60 − 4.60i)13-s + (−0.691 + 1.19i)14-s + (−2.03 + 3.44i)16-s − 0.490i·17-s + (−0.754 − 0.438i)18-s + (−4.54 + 4.54i)19-s + ⋯
L(s)  = 1  + (0.502 − 0.864i)2-s + (−0.630 − 0.630i)3-s + (−0.495 − 0.868i)4-s + (−0.861 + 0.228i)6-s − 0.368·7-s + (−0.999 − 0.00714i)8-s − 0.205i·9-s + (0.424 + 0.424i)11-s + (−0.234 + 0.859i)12-s + (−1.27 − 1.27i)13-s + (−0.184 + 0.318i)14-s + (−0.508 + 0.861i)16-s − 0.118i·17-s + (−0.177 − 0.103i)18-s + (−1.04 + 1.04i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192912 + 0.819891i\)
\(L(\frac12)\) \(\approx\) \(0.192912 + 0.819891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.710 + 1.22i)T \)
5 \( 1 \)
good3 \( 1 + (1.09 + 1.09i)T + 3iT^{2} \)
7 \( 1 + 0.973T + 7T^{2} \)
11 \( 1 + (-1.40 - 1.40i)T + 11iT^{2} \)
13 \( 1 + (4.60 + 4.60i)T + 13iT^{2} \)
17 \( 1 + 0.490iT - 17T^{2} \)
19 \( 1 + (4.54 - 4.54i)T - 19iT^{2} \)
23 \( 1 - 1.94T + 23T^{2} \)
29 \( 1 + (-3.74 + 3.74i)T - 29iT^{2} \)
31 \( 1 - 4.29T + 31T^{2} \)
37 \( 1 + (-4.55 + 4.55i)T - 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (5.61 - 5.61i)T - 53iT^{2} \)
59 \( 1 + (8.44 + 8.44i)T + 59iT^{2} \)
61 \( 1 + (-3.01 + 3.01i)T - 61iT^{2} \)
67 \( 1 + (7.07 + 7.07i)T + 67iT^{2} \)
71 \( 1 - 0.897iT - 71T^{2} \)
73 \( 1 - 9.71T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (-0.815 - 0.815i)T + 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 - 7.54iT - 97T^{2} \)
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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

−10.83504484511315382789761443908, −10.07697364347485464377998035793, −9.272885873479294788373008597978, −7.894688544809239092691037068091, −6.66665850070132362579807743558, −5.88158189933830516340111782608, −4.83060500501486187006797390656, −3.54640388677372587554501704464, −2.18411131744582461519217189198, −0.49090464574293727407605157716, 2.82459334574475381742898305795, 4.50207584059076795760565954701, 4.74806208904479348888386859645, 6.20007502593321581061277183377, 6.77540914654573642336207081596, 7.976335046180691614333500428801, 9.091414907665439740404367079733, 9.804714195197676884726750740604, 11.07144331325946336267106923287, 11.77710911590087578978113065893

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