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Properties

Label 1-293-293.90-r0-0-0
Degree $1$
Conductor $293$
Sign $0.900 + 0.435i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
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  + (−0.474 − 0.880i)2-s + (−0.744 + 0.668i)3-s + (−0.548 + 0.835i)4-s + (0.714 + 0.699i)5-s + (0.941 + 0.337i)6-s + (0.985 − 0.171i)7-s + (0.996 + 0.0859i)8-s + (0.107 − 0.994i)9-s + (0.276 − 0.961i)10-s + (0.357 + 0.933i)11-s + (−0.150 − 0.988i)12-s + (0.772 + 0.635i)13-s + (−0.618 − 0.785i)14-s + (−0.999 − 0.0430i)15-s + (−0.397 − 0.917i)16-s + (−0.474 − 0.880i)17-s + ⋯
L(s)  = 1  + (−0.474 − 0.880i)2-s + (−0.744 + 0.668i)3-s + (−0.548 + 0.835i)4-s + (0.714 + 0.699i)5-s + (0.941 + 0.337i)6-s + (0.985 − 0.171i)7-s + (0.996 + 0.0859i)8-s + (0.107 − 0.994i)9-s + (0.276 − 0.961i)10-s + (0.357 + 0.933i)11-s + (−0.150 − 0.988i)12-s + (0.772 + 0.635i)13-s + (−0.618 − 0.785i)14-s + (−0.999 − 0.0430i)15-s + (−0.397 − 0.917i)16-s + (−0.474 − 0.880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.900 + 0.435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9134660061 + 0.2093001789i\)
\(L(\frac12)\) \(\approx\) \(0.9134660061 + 0.2093001789i\)
\(L(1)\) \(\approx\) \(0.8360360565 + 0.02299325106i\)
\(L(1)\) \(\approx\) \(0.8360360565 + 0.02299325106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.474 - 0.880i)T \)
3 \( 1 + (-0.744 + 0.668i)T \)
5 \( 1 + (0.714 + 0.699i)T \)
7 \( 1 + (0.985 - 0.171i)T \)
11 \( 1 + (0.357 + 0.933i)T \)
13 \( 1 + (0.772 + 0.635i)T \)
17 \( 1 + (-0.474 - 0.880i)T \)
19 \( 1 + (-0.999 - 0.0430i)T \)
23 \( 1 + (0.941 - 0.337i)T \)
29 \( 1 + (-0.847 - 0.530i)T \)
31 \( 1 + (0.714 - 0.699i)T \)
37 \( 1 + (-0.683 + 0.729i)T \)
41 \( 1 + (0.869 + 0.493i)T \)
43 \( 1 + (0.966 + 0.255i)T \)
47 \( 1 + (0.651 + 0.758i)T \)
53 \( 1 + (0.357 + 0.933i)T \)
59 \( 1 + (-0.954 - 0.296i)T \)
61 \( 1 + (-0.798 - 0.601i)T \)
67 \( 1 + (0.823 - 0.566i)T \)
71 \( 1 + (-0.890 + 0.455i)T \)
73 \( 1 + (-0.548 - 0.835i)T \)
79 \( 1 + (-0.798 - 0.601i)T \)
83 \( 1 + (0.584 + 0.811i)T \)
89 \( 1 + (-0.798 + 0.601i)T \)
97 \( 1 + (-0.234 + 0.972i)T \)
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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

−25.08233787893968638721233197970, −24.52006233717431841938393790580, −23.910778093521911791803093344025, −23.05968214430418597468126045781, −21.88842357184120913646383372158, −21.031069166887806435589805131753, −19.611489105313728731795322801394, −18.73960269786541493596289123793, −17.74661428174850938900456837003, −17.317934466788178954681090779471, −16.56100399952331416412213226917, −15.48956656848040007789708431168, −14.2699098348964996310853576124, −13.42466193445120419538008042454, −12.61043129773171475978722902274, −11.07731473573070676480451244186, −10.53705847524267148976201503653, −8.758427753699318186808580008676, −8.48071472915565356813833235547, −7.16429863716823774098062033259, −5.93022742102860901756000624228, −5.57930829480529741180085090275, −4.387085089272987486560526246022, −1.846222843629410761475803367195, −0.96619694436665066056273113735, 1.37617151150937707190256545666, 2.54670930449870690751370266410, 4.10599326000032274745656274862, 4.7881131893738443106633879927, 6.29686570414780980016522283941, 7.38921464203183949807606393235, 8.96516393278866262319275045568, 9.62637957812282208362470903485, 10.79555581444276546432507206461, 11.11800760769166153435054828958, 12.123112252084073896437842080019, 13.38557553174511913441808741551, 14.418429785042759273005003333976, 15.418368211761706393162880962514, 16.92510541875116958436471414310, 17.366396286317679697600536188416, 18.17403031009695515361574641000, 18.94179808710710113638378506558, 20.580174691894571239155824870206, 20.89389195331295107313310706541, 21.75404832031380971361979241867, 22.658072695343783349404448923484, 23.23500845461255797598548371610, 24.75701397922141691714671581463, 25.97424747641590390209899403676

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