↑

Properties

Label 1-403-403.159-r0-0-0
Degree $1$
Conductor $403$
Sign $0.0210 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.104 − 0.994i)10-s + (0.669 + 0.743i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)14-s + (0.913 − 0.406i)15-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.104 − 0.994i)10-s + (0.669 + 0.743i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)14-s + (0.913 − 0.406i)15-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0210 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0210 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.0210 - 0.999i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.0210 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266729711 - 1.240358741i\)
\(L(\frac12)\) \(\approx\) \(1.266729711 - 1.240358741i\)
\(L(1)\) \(\approx\) \(1.178308648 - 0.7473192362i\)
\(L(1)\) \(\approx\) \(1.178308648 - 0.7473192362i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−24.681478502756814700601673848422, −24.11454024972583422474454325772, −22.72182819556423988305666963014, −21.963434062430635132768582164527, −21.42237086857882235865633256889, −20.05234806225318214219489764213, −19.30692108441628194060421814159, −18.44274950900304157747207701866, −17.39445932544301001881714432703, −16.38793281365104564807943050390, −16.0080166452391728101066218822, −14.70642701475082264068345048611, −14.08811325498021916321181588157, −13.44012912932532565928725555697, −12.58330081423419621844377508699, −10.60560212015700915715130660097, −9.713445702229896913453756368343, −9.23784861708828132708681432408, −8.28412221007846099949031754886, −7.14258549026336196161716067151, −6.19320164564302730924392514110, −5.31770955586398517819982220146, −3.918871854702501622064089891252, −3.10105530157529596145188751892, −1.399943572814480933274977576159, 1.21794183221232948100825075388, 2.24773335370837557348620086585, 3.07298339305516386957846594175, 4.099814882508932835236828100978, 5.56520754138941751137230377209, 6.74490319073985802623392427373, 7.91452878354386941025084083685, 9.19608677046563242207371424640, 9.607881069469965854022463668900, 10.24688677225129198171408156611, 12.024502941591911607097710375776, 12.42625936400233847106923446422, 13.63188185610228124965769102128, 13.89659977881230089426281119741, 14.9990586605711412250919011371, 16.383786871423137045134994500, 17.50309405740838759698343904831, 18.3292895291839211109174023044, 18.962955168554960593521110970083, 19.93435741181882290297611026369, 20.52839223016607644194023940839, 21.31688157638987130182010266303, 22.36809427058295769727470149300, 22.8540858108593394331435158549, 24.337369362404849266112288971820

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