Properties

Label 2-20e2-25.4-c1-0-0
Degree $2$
Conductor $400$
Sign $-0.940 + 0.340i$
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  + (−1.29 + 1.77i)3-s + (−1.22 + 1.87i)5-s − 0.992i·7-s + (−0.566 − 1.74i)9-s + (−0.618 + 1.90i)11-s + (−3.20 + 1.04i)13-s + (−1.74 − 4.59i)15-s + (−1.70 − 2.34i)17-s + (2.09 − 1.51i)19-s + (1.76 + 1.28i)21-s + (−4.32 − 1.40i)23-s + (−1.99 − 4.58i)25-s + (−2.43 − 0.792i)27-s + (−4.35 − 3.16i)29-s + (0.110 − 0.0802i)31-s + ⋯
L(s)  = 1  + (−0.746 + 1.02i)3-s + (−0.548 + 0.836i)5-s − 0.375i·7-s + (−0.188 − 0.581i)9-s + (−0.186 + 0.573i)11-s + (−0.889 + 0.289i)13-s + (−0.449 − 1.18i)15-s + (−0.412 − 0.567i)17-s + (0.479 − 0.348i)19-s + (0.385 + 0.279i)21-s + (−0.902 − 0.293i)23-s + (−0.399 − 0.916i)25-s + (−0.469 − 0.152i)27-s + (−0.808 − 0.587i)29-s + (0.0198 − 0.0144i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.940 + 0.340i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0670560 - 0.381859i\)
\(L(\frac12)\) \(\approx\) \(0.0670560 - 0.381859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.87i)T \)
good3 \( 1 + (1.29 - 1.77i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.992iT - 7T^{2} \)
11 \( 1 + (0.618 - 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.20 - 1.04i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.70 + 2.34i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.09 + 1.51i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.32 + 1.40i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.35 + 3.16i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.110 + 0.0802i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.04 + 0.664i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.66 - 8.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.64iT - 43T^{2} \)
47 \( 1 + (5.83 - 8.03i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.44 - 6.12i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.51 - 4.67i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.855 - 2.63i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.28 - 1.76i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-7.80 - 5.66i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.737 + 0.239i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.8 + 9.31i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.04 - 1.43i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.48 - 13.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.0 - 13.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.44916395221387384690570896078, −10.97003047166596017277913925180, −9.958764964330219759357040014431, −9.546507376545711185547109653474, −7.86840081321845244176870214518, −7.13668135855490138879689174071, −6.00917085489310723327990317619, −4.73280706094995315019557887021, −4.12613989615861875711192356743, −2.64269344687297629148693581014, 0.26762875190875854338943461332, 1.85922797767435635552452760230, 3.72111527946526329282785604665, 5.19964641760533392974552137586, 5.85763629040742070401894254350, 7.07146613297705055865471166088, 7.87472212802668502435082321531, 8.742801478852504646787374076036, 9.857594893018519624328275796109, 11.12646650717022736710205706354

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