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Properties

Label 1-403-403.113-r0-0-0
Degree $1$
Conductor $403$
Sign $0.884 + 0.466i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (0.913 + 0.406i)2-s + (0.913 − 0.406i)3-s + (0.669 + 0.743i)4-s + (−0.5 + 0.866i)5-s + 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.809 + 0.587i)10-s + (0.309 − 0.951i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (−0.104 + 0.994i)15-s + (−0.104 + 0.994i)16-s + (0.309 + 0.951i)17-s + (0.913 − 0.406i)18-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.913 − 0.406i)3-s + (0.669 + 0.743i)4-s + (−0.5 + 0.866i)5-s + 6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.809 + 0.587i)10-s + (0.309 − 0.951i)11-s + (0.913 + 0.406i)12-s + (0.669 − 0.743i)14-s + (−0.104 + 0.994i)15-s + (−0.104 + 0.994i)16-s + (0.309 + 0.951i)17-s + (0.913 − 0.406i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.901571570 + 0.7185601231i\)
\(L(\frac12)\) \(\approx\) \(2.901571570 + 0.7185601231i\)
\(L(1)\) \(\approx\) \(2.198100679 + 0.4042562908i\)
\(L(1)\) \(\approx\) \(2.198100679 + 0.4042562908i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.34096704383924288556854703697, −23.384290701182210762970522473349, −22.4695089683480944223675813344, −21.369323343475946835436888832501, −21.00095839971448627142663469835, −20.05900358229532142216468694666, −19.49493152753191036503304454831, −18.5830634078923146993220526239, −17.103197823126113463727107665818, −15.72640652108282662634603867389, −15.495456859277829707723807378297, −14.59427781563386727183770516585, −13.64396120595509937689898995206, −12.66744509832922060828588027050, −12.04025928411864557390112267470, −11.0332822668453065058412267395, −9.68473602890547238344964562040, −9.06387330746306192785096664214, −7.93897271466734917807575344611, −6.83199407255176468441377287317, −5.167062750791989189240913315005, −4.745031278093786434629595313954, −3.65888890440361498037615418914, −2.53990178620669032248861823786, −1.57392174409494925921686419069, 1.620561599719052296166052025054, 3.0758305575664054776192223352, 3.60813761664430046868078043798, 4.594221362789713727045110301551, 6.40505045874613745371572008709, 6.77906170258884216644961242289, 8.05627447828703057662839901896, 8.34045340318907536237840581991, 10.25982456108635979964869848356, 11.06783503618840738897788165975, 12.15332937194380159116203638709, 13.143052054639947092849676455365, 13.98538459678783006609299300984, 14.584952473030678581595389479528, 15.18493445248236022919522322156, 16.37680596078513269945824543450, 17.230803679265768269520137657521, 18.50502487711999485754003519655, 19.38996972092622532195089677102, 20.07933072312059797118989583142, 21.11750118010990138726291629514, 21.75536291778660825596269061456, 22.98686455929282909192755897929, 23.57998838301403201614412791856, 24.26037688037822982479218369341

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