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Properties

Label 2-403-31.4-c1-0-28
Degree $2$
Conductor $403$
Sign $-0.749 + 0.662i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + (−0.288 − 0.888i)2-s + (0.813 − 2.50i)3-s + (0.911 − 0.662i)4-s + 1.99·5-s − 2.46·6-s + (0.592 − 0.430i)7-s + (−2.36 − 1.71i)8-s + (−3.18 − 2.31i)9-s + (−0.577 − 1.77i)10-s + (−2.80 + 2.03i)11-s + (−0.916 − 2.82i)12-s + (−0.309 + 0.951i)13-s + (−0.553 − 0.402i)14-s + (1.62 − 5.00i)15-s + (−0.147 + 0.453i)16-s + (5.84 + 4.24i)17-s + ⋯
L(s)  = 1  + (−0.204 − 0.628i)2-s + (0.469 − 1.44i)3-s + (0.455 − 0.331i)4-s + 0.893·5-s − 1.00·6-s + (0.223 − 0.162i)7-s + (−0.835 − 0.607i)8-s + (−1.06 − 0.770i)9-s + (−0.182 − 0.561i)10-s + (−0.846 + 0.614i)11-s + (−0.264 − 0.814i)12-s + (−0.0857 + 0.263i)13-s + (−0.147 − 0.107i)14-s + (0.419 − 1.29i)15-s + (−0.0368 + 0.113i)16-s + (1.41 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636501 - 1.68139i\)
\(L(\frac12)\) \(\approx\) \(0.636501 - 1.68139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (3.84 + 4.02i)T \)
good2 \( 1 + (0.288 + 0.888i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.813 + 2.50i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 1.99T + 5T^{2} \)
7 \( 1 + (-0.592 + 0.430i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.80 - 2.03i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (-5.84 - 4.24i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.108 - 0.333i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.42 - 3.21i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.974 - 3.00i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 5.79T + 37T^{2} \)
41 \( 1 + (0.429 + 1.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.794 - 2.44i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.59 - 4.90i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.69 + 1.23i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.17 + 6.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 4.26T + 61T^{2} \)
67 \( 1 + 5.16T + 67T^{2} \)
71 \( 1 + (-3.45 - 2.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-10.0 + 7.28i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.24 + 6.72i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.81 + 5.57i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.28 + 5.29i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (10.5 - 7.64i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.87858878368268185504631228496, −10.06445384724681644655963043280, −9.246829605025083560059246596902, −7.959472095558752110190590187777, −7.25333213261289794177005413140, −6.26851382944063873568650169070, −5.39263597756589648284399376242, −3.20958213687082547888603682098, −2.05483862979715367284409906385, −1.38884073876296424525753842881, 2.59275588888393827268770969154, 3.41129613995814228843216824869, 5.15774568449689466276690796226, 5.58303411183964565171622676434, 7.00564608767052147401004547099, 8.137113119091190128657537046016, 8.844062488267488710171366700182, 9.766102249015276355181488587066, 10.45077386990704475906532466289, 11.33762303276165490528793954485

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