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Properties

Label 2-403-31.2-c1-0-5
Degree $2$
Conductor $403$
Sign $-0.502 - 0.864i$
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  + (1.04 + 0.757i)2-s + (−0.137 + 0.0998i)3-s + (−0.105 − 0.324i)4-s − 3.49·5-s − 0.218·6-s + (1.29 + 3.97i)7-s + (0.931 − 2.86i)8-s + (−0.918 + 2.82i)9-s + (−3.64 − 2.64i)10-s + (1.71 + 5.28i)11-s + (0.0468 + 0.0340i)12-s + (0.809 − 0.587i)13-s + (−1.66 + 5.12i)14-s + (0.480 − 0.349i)15-s + (2.59 − 1.88i)16-s + (−1.09 + 3.35i)17-s + ⋯
L(s)  = 1  + (0.736 + 0.535i)2-s + (−0.0793 + 0.0576i)3-s + (−0.0526 − 0.162i)4-s − 1.56·5-s − 0.0893·6-s + (0.488 + 1.50i)7-s + (0.329 − 1.01i)8-s + (−0.306 + 0.941i)9-s + (−1.15 − 0.836i)10-s + (0.517 + 1.59i)11-s + (0.0135 + 0.00983i)12-s + (0.224 − 0.163i)13-s + (−0.445 + 1.36i)14-s + (0.124 − 0.0901i)15-s + (0.647 − 0.470i)16-s + (−0.264 + 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639551 + 1.11137i\)
\(L(\frac12)\) \(\approx\) \(0.639551 + 1.11137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.86 + 4.77i)T \)
good2 \( 1 + (-1.04 - 0.757i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.137 - 0.0998i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 3.49T + 5T^{2} \)
7 \( 1 + (-1.29 - 3.97i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.71 - 5.28i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (1.09 - 3.35i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.00 + 3.63i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.01 - 3.12i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.47 - 1.07i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 9.72T + 37T^{2} \)
41 \( 1 + (-6.55 - 4.75i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.59 + 3.33i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-3.26 + 2.37i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.32 + 4.06i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.15 + 5.19i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 2.09T + 61T^{2} \)
67 \( 1 - 6.69T + 67T^{2} \)
71 \( 1 + (2.93 - 9.03i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.553 + 1.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.79 - 14.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.16 - 5.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.71 + 5.27i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.14 - 3.52i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.61148364815691981423667988869, −10.99201888202357833695830417201, −9.684415417838661319603931352581, −8.570083822688474684837185064444, −7.80331003614159704543879292417, −6.79614927207535048644531394919, −5.65500948317443141438808884429, −4.67736919875095447234911881224, −4.07709792018981287387230754811, −2.20095021673149159482288970318, 0.69443382588243529051871587988, 3.20318919114493463735056510924, 3.95442678913543253382432684309, 4.43823034168060489919840281363, 6.15459673419859866432887995737, 7.32959972035836070141661693687, 8.178077021493225721819557939731, 8.834424443607898854712926262108, 10.67327615837277713867784408782, 11.20064339069435846496463157149

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