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Properties

Label 1-293-293.56-r0-0-0
Degree $1$
Conductor $293$
Sign $0.822 - 0.568i$
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  + (−0.0645 − 0.997i)2-s + (0.651 + 0.758i)3-s + (−0.991 + 0.128i)4-s + (0.985 − 0.171i)5-s + (0.714 − 0.699i)6-s + (−0.925 − 0.377i)7-s + (0.192 + 0.981i)8-s + (−0.150 + 0.988i)9-s + (−0.234 − 0.972i)10-s + (0.908 + 0.417i)11-s + (−0.744 − 0.668i)12-s + (0.0215 − 0.999i)13-s + (−0.317 + 0.948i)14-s + (0.772 + 0.635i)15-s + (0.966 − 0.255i)16-s + (−0.0645 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.0645 − 0.997i)2-s + (0.651 + 0.758i)3-s + (−0.991 + 0.128i)4-s + (0.985 − 0.171i)5-s + (0.714 − 0.699i)6-s + (−0.925 − 0.377i)7-s + (0.192 + 0.981i)8-s + (−0.150 + 0.988i)9-s + (−0.234 − 0.972i)10-s + (0.908 + 0.417i)11-s + (−0.744 − 0.668i)12-s + (0.0215 − 0.999i)13-s + (−0.317 + 0.948i)14-s + (0.772 + 0.635i)15-s + (0.966 − 0.255i)16-s + (−0.0645 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.462543834 - 0.4565442683i\)
\(L(\frac12)\) \(\approx\) \(1.462543834 - 0.4565442683i\)
\(L(1)\) \(\approx\) \(1.242882327 - 0.3282497030i\)
\(L(1)\) \(\approx\) \(1.242882327 - 0.3282497030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.0645 - 0.997i)T \)
3 \( 1 + (0.651 + 0.758i)T \)
5 \( 1 + (0.985 - 0.171i)T \)
7 \( 1 + (-0.925 - 0.377i)T \)
11 \( 1 + (0.908 + 0.417i)T \)
13 \( 1 + (0.0215 - 0.999i)T \)
17 \( 1 + (-0.0645 - 0.997i)T \)
19 \( 1 + (0.772 + 0.635i)T \)
23 \( 1 + (0.714 + 0.699i)T \)
29 \( 1 + (-0.890 + 0.455i)T \)
31 \( 1 + (0.985 + 0.171i)T \)
37 \( 1 + (0.869 - 0.493i)T \)
41 \( 1 + (-0.397 + 0.917i)T \)
43 \( 1 + (-0.548 - 0.835i)T \)
47 \( 1 + (0.357 + 0.933i)T \)
53 \( 1 + (0.908 + 0.417i)T \)
59 \( 1 + (0.107 - 0.994i)T \)
61 \( 1 + (-0.618 - 0.785i)T \)
67 \( 1 + (-0.976 + 0.213i)T \)
71 \( 1 + (0.276 - 0.961i)T \)
73 \( 1 + (-0.991 - 0.128i)T \)
79 \( 1 + (-0.618 - 0.785i)T \)
83 \( 1 + (-0.847 + 0.530i)T \)
89 \( 1 + (-0.618 + 0.785i)T \)
97 \( 1 + (-0.798 - 0.601i)T \)
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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

−25.53957093477825962240936508827, −24.62721275417563541402933348950, −24.238498846944385632824942559322, −22.940046366977530718534881392030, −22.09873010124134269203797133089, −21.2761008248654817125238837471, −19.74702723597077248519157395880, −18.946231692091939502854161196008, −18.35129738061621662216674625920, −17.2004835475722060695933655017, −16.60579631832594883516490879593, −15.21870159038244998916295407087, −14.48401628404988259099665518809, −13.53352651585846412215012070538, −13.10405263744639442991127437412, −11.84294079602462144560619637875, −9.99495667199659413799630106865, −9.14129070272222531358702909660, −8.63798930923464574469698057402, −7.071620642212105055094808446438, −6.48529633334172830410965175069, −5.75824749235439686751126534134, −4.037599043733927102946421560910, −2.76564605816394629155596970439, −1.28674888825918193319788388245, 1.32558336947291899011242861342, 2.75837764904980635023724075051, 3.46298538850015618972878098078, 4.70447098427267356564908049041, 5.73809079438873831136173702875, 7.42811528093948598739469492081, 8.85773490881664523192418534767, 9.65075727858917385272762809956, 9.98269774541275559123270475095, 11.11452056628403011551424757518, 12.45774332822023299385956095014, 13.4167906212911317075755447327, 13.97404086236575478899778744957, 15.04578997068026401527597492753, 16.403417358380868739221639819310, 17.18459249111026873065394882618, 18.23389161690753906082201749017, 19.33494139699733400355776523897, 20.27895698431545303940835039775, 20.54671414123982020063362780559, 21.735710852790273432357611662622, 22.397062203076803080572611293485, 22.99918308211281098341506903266, 24.97279406509340464872007795322, 25.36799179073870586514815180208

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