Properties

Label 2-403-403.185-c0-0-1
Degree $2$
Conductor $403$
Sign $-0.252 + 0.967i$
Analytic cond. $0.201123$
Root an. cond. $0.448467$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.866 − 0.499i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.866 + 0.5i)11-s + i·13-s − 0.999·14-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (0.866 + 0.499i)22-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.499i)24-s − 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.866 − 0.499i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.866 + 0.5i)11-s + i·13-s − 0.999·14-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (0.866 + 0.499i)22-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.499i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(0.201123\)
Root analytic conductor: \(0.448467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :0),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8506803330\)
\(L(\frac12)\) \(\approx\) \(0.8506803330\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - iT \)
31 \( 1 + iT \)
good2 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.26796448480203553455967327815, −10.12367844768355126479479256389, −9.703931011239355378131816532714, −8.419997929838757551138900962459, −7.80136424471324208933566855871, −6.80157891457166250840806925404, −5.34181723325164253808521418908, −3.87440947248701690403815744092, −2.53886931670367344245009654332, −1.60641294567980630786349282588, 2.66549256429074363496068972055, 3.48762579600775443326674568583, 5.32693212338431208144727800978, 5.98085371872081457354262675499, 7.50834843972665177469714238890, 8.216775647198376174303104780645, 8.640817716924347589576077843595, 9.675764230718966348357525302388, 10.50276671636804297753257264728, 11.97706412431208658778877458540

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