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Properties

Label 2-403-31.4-c1-0-3
Degree $2$
Conductor $403$
Sign $0.965 + 0.258i$
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  + (−0.809 − 2.48i)2-s + (0.190 − 0.587i)3-s + (−3.92 + 2.85i)4-s + 0.381·5-s − 1.61·6-s + (−3.92 + 2.85i)7-s + (6.04 + 4.39i)8-s + (2.11 + 1.53i)9-s + (−0.309 − 0.951i)10-s + (0.190 − 0.138i)11-s + (0.927 + 2.85i)12-s + (−0.309 + 0.951i)13-s + (10.2 + 7.46i)14-s + (0.0729 − 0.224i)15-s + (3.04 − 9.37i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.572 − 1.76i)2-s + (0.110 − 0.339i)3-s + (−1.96 + 1.42i)4-s + 0.170·5-s − 0.660·6-s + (−1.48 + 1.07i)7-s + (2.13 + 1.55i)8-s + (0.706 + 0.512i)9-s + (−0.0977 − 0.300i)10-s + (0.0575 − 0.0418i)11-s + (0.267 + 0.823i)12-s + (−0.0857 + 0.263i)13-s + (2.74 + 1.99i)14-s + (0.0188 − 0.0579i)15-s + (0.761 − 2.34i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.965 + 0.258i$
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: \((2,\ 403,\ (\ :1/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620433 - 0.0816203i\)
\(L(\frac12)\) \(\approx\) \(0.620433 - 0.0816203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 5.42i)T \)
good2 \( 1 + (0.809 + 2.48i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 0.381T + 5T^{2} \)
7 \( 1 + (3.92 - 2.85i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.190 + 0.138i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.854 - 2.62i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-5.23 - 3.80i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.263 - 0.812i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 5.38T + 37T^{2} \)
41 \( 1 + (0.427 + 1.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.73 - 5.34i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.66 - 11.2i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.3 + 7.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.11 - 3.44i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 + 3.38T + 67T^{2} \)
71 \( 1 + (-3.85 - 2.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.66 + 4.84i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.04 - 3.66i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.38 + 7.33i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.66 - 4.84i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (9.70 - 7.05i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.24380925434652411978889530847, −10.21129010978354829283709618726, −9.547749995991426015680414410087, −9.004713780974783607799268105086, −7.88071922852222241705357807336, −6.60100973353018153839784316542, −5.13742340211316390315862135694, −3.64934481706973569516732077371, −2.74659772421804209870515410768, −1.62248128556189786368238006893, 0.50835511789390250391212272553, 3.61391520066988911979484920303, 4.64332873763314535047850803142, 5.94498452853961682106333182453, 6.87119296854443606383149554927, 7.18963157897997378021673073603, 8.490397152033507893209843194989, 9.548347097629416529313170555735, 9.802411595044232003212594862802, 10.74110979795690992779947709222

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