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Properties

Label 2-403-31.4-c1-0-12
Degree $2$
Conductor $403$
Sign $0.817 - 0.576i$
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  + (−0.00310 − 0.00954i)2-s + (−0.517 + 1.59i)3-s + (1.61 − 1.17i)4-s + 1.33·5-s + 0.0167·6-s + (−0.925 + 0.672i)7-s + (−0.0324 − 0.0235i)8-s + (0.161 + 0.117i)9-s + (−0.00414 − 0.0127i)10-s + (1.99 − 1.45i)11-s + (1.03 + 3.18i)12-s + (−0.309 + 0.951i)13-s + (0.00928 + 0.00674i)14-s + (−0.691 + 2.12i)15-s + (1.23 − 3.80i)16-s + (3.66 + 2.66i)17-s + ⋯
L(s)  = 1  + (−0.00219 − 0.00674i)2-s + (−0.298 + 0.918i)3-s + (0.808 − 0.587i)4-s + 0.598·5-s + 0.00685·6-s + (−0.349 + 0.254i)7-s + (−0.0114 − 0.00834i)8-s + (0.0539 + 0.0392i)9-s + (−0.00131 − 0.00403i)10-s + (0.603 − 0.438i)11-s + (0.298 + 0.918i)12-s + (−0.0857 + 0.263i)13-s + (0.00248 + 0.00180i)14-s + (−0.178 + 0.549i)15-s + (0.308 − 0.950i)16-s + (0.888 + 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.817 - 0.576i$
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: \((2,\ 403,\ (\ :1/2),\ 0.817 - 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58566 + 0.502579i\)
\(L(\frac12)\) \(\approx\) \(1.58566 + 0.502579i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.37 - 4.42i)T \)
good2 \( 1 + (0.00310 + 0.00954i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.517 - 1.59i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 1.33T + 5T^{2} \)
7 \( 1 + (0.925 - 0.672i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.99 + 1.45i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (-3.66 - 2.66i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.291 + 0.897i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.55 + 1.13i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.766 - 2.35i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 8.90T + 37T^{2} \)
41 \( 1 + (3.17 + 9.76i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.29 - 4.00i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.675 - 2.07i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.85 + 5.70i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.37 + 4.22i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 5.99T + 61T^{2} \)
67 \( 1 + 8.82T + 67T^{2} \)
71 \( 1 + (12.3 + 9.00i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (7.08 - 5.14i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (13.4 + 9.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.15 + 9.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.01 + 6.54i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-7.49 + 5.44i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.19376118312070623132369290647, −10.27676053665249883071473223742, −9.870536607270387585612334816078, −8.924519439488171317464136256168, −7.50440648295189221164011444961, −6.27190379338114128270623661896, −5.75769572919080695117371574446, −4.58020141286626756987570379696, −3.20237853451708090843343041204, −1.66910969998947471852620440719, 1.40001271740083185435037343694, 2.67640373182015016456833185076, 4.06986417574733274186906765449, 5.84878413779741830594303424843, 6.48865435931066720633569947963, 7.37502256752758989502620186302, 8.017103959078159843395692558133, 9.564954058366384894927389391201, 10.16578916980619315352633405028, 11.67093887073750323150785755892

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