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Properties

Label 2-403-31.16-c1-0-30
Degree $2$
Conductor $403$
Sign $-0.912 + 0.408i$
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  + (2.05 − 1.49i)2-s + (−1.94 − 1.41i)3-s + (1.37 − 4.24i)4-s + 1.44·5-s − 6.10·6-s + (0.616 − 1.89i)7-s + (−1.93 − 5.95i)8-s + (0.854 + 2.62i)9-s + (2.97 − 2.15i)10-s + (−0.941 + 2.89i)11-s + (−8.66 + 6.29i)12-s + (0.809 + 0.587i)13-s + (−1.56 − 4.82i)14-s + (−2.80 − 2.03i)15-s + (−5.65 − 4.10i)16-s + (0.846 + 2.60i)17-s + ⋯
L(s)  = 1  + (1.45 − 1.05i)2-s + (−1.12 − 0.814i)3-s + (0.689 − 2.12i)4-s + 0.646·5-s − 2.49·6-s + (0.233 − 0.717i)7-s + (−0.683 − 2.10i)8-s + (0.284 + 0.876i)9-s + (0.939 − 0.682i)10-s + (−0.283 + 0.874i)11-s + (−2.50 + 1.81i)12-s + (0.224 + 0.163i)13-s + (−0.419 − 1.29i)14-s + (−0.724 − 0.526i)15-s + (−1.41 − 1.02i)16-s + (0.205 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.912 + 0.408i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477839 - 2.24011i\)
\(L(\frac12)\) \(\approx\) \(0.477839 - 2.24011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (5.11 - 2.19i)T \)
good2 \( 1 + (-2.05 + 1.49i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.94 + 1.41i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 + (-0.616 + 1.89i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.941 - 2.89i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (-0.846 - 2.60i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.156 - 0.113i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.579 + 1.78i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.38 + 4.63i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 7.58T + 37T^{2} \)
41 \( 1 + (-6.48 + 4.70i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (4.59 - 3.33i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-3.83 - 2.78i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.845 + 2.60i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.58 + 5.50i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 2.54T + 61T^{2} \)
67 \( 1 - 9.61T + 67T^{2} \)
71 \( 1 + (3.60 + 11.0i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.47 - 10.6i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.41 + 4.36i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.975 - 0.708i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (5.04 - 15.5i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.37 - 10.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.07586241380524352841169300835, −10.51326108935585861412791420155, −9.649031047657457569622260901137, −7.68718683213960945908669266120, −6.51416881587981492667507526097, −5.91905070129823985517519546754, −4.94207718496964362912432172186, −4.00435841852346271444406577115, −2.29725242238346956754103001098, −1.22212848106144312048498453396, 2.87215165216740198008289743107, 4.20625873184832313936218728860, 5.22676758073681004735833534436, 5.70289555971433599391221499239, 6.29813143084175248283470030004, 7.60683441508504038057233170467, 8.764788302877526010052357295148, 9.981071370041678695984578539941, 11.12593140159748547042338669339, 11.73435826173288697080953354726

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