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Properties

Label 2-403-31.2-c1-0-9
Degree $2$
Conductor $403$
Sign $0.569 + 0.822i$
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 − 0.587i)2-s + (−1.61 + 1.17i)3-s + (−0.309 − 0.951i)4-s − 3.23·5-s + 2·6-s + (0.809 + 2.48i)7-s + (−0.927 + 2.85i)8-s + (0.309 − 0.951i)9-s + (2.61 + 1.90i)10-s + (−0.5 − 1.53i)11-s + (1.61 + 1.17i)12-s + (0.809 − 0.587i)13-s + (0.809 − 2.48i)14-s + (5.23 − 3.80i)15-s + (0.809 − 0.587i)16-s + (1.42 − 4.39i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.934 + 0.678i)3-s + (−0.154 − 0.475i)4-s − 1.44·5-s + 0.816·6-s + (0.305 + 0.941i)7-s + (−0.327 + 1.00i)8-s + (0.103 − 0.317i)9-s + (0.827 + 0.601i)10-s + (−0.150 − 0.463i)11-s + (0.467 + 0.339i)12-s + (0.224 − 0.163i)13-s + (0.216 − 0.665i)14-s + (1.35 − 0.982i)15-s + (0.202 − 0.146i)16-s + (0.346 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.569 + 0.822i$
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: \((2,\ 403,\ (\ :1/2),\ 0.569 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356081 - 0.186602i\)
\(L(\frac12)\) \(\approx\) \(0.356081 - 0.186602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 - 4.53i)T \)
good2 \( 1 + (0.809 + 0.587i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.61 - 1.17i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
7 \( 1 + (-0.809 - 2.48i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.5 + 1.53i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-1.42 + 4.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.61 + 4.97i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.5 + 1.81i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-7.85 - 5.70i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-9.59 + 6.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.618 - 1.90i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.54 - 4.75i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 6.61T + 67T^{2} \)
71 \( 1 + (-0.5 + 1.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.38 + 7.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.76 + 11.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.92 - 1.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.23 + 3.80i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.52 - 7.77i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.99374870989098124990675802593, −10.56516394083547493041828570450, −9.346432287401098808485239665819, −8.558661168252559607586718167521, −7.68055329408569500775825804039, −6.08740666144220429149862523119, −5.20208205010695879205138925116, −4.40653707215554206586527704777, −2.77272096739335599497878422339, −0.51631928000856545030200929995, 0.921693494818302228843595773958, 3.62073719996397896498502552822, 4.35550831915590512595117216231, 5.97130090661033623977551299443, 7.17483196850584538117256010088, 7.49333102890119888557642701961, 8.259166719887154597644232276325, 9.447327556223314638665792274913, 10.77728902739193453955836531432, 11.39056283954192555761509309788

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