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Properties

Label 1-2340-2340.7-r1-0-0
Degree $1$
Conductor $2340$
Sign $-0.992 + 0.118i$
Analytic cond. $251.467$
Root an. cond. $251.467$
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  − 7-s + (−0.866 − 0.5i)11-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s − i·23-s + (0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.5 − 0.866i)37-s + i·41-s − i·43-s + (0.5 + 0.866i)47-s + 49-s − i·53-s + (−0.866 + 0.5i)59-s + 61-s + ⋯
L(s)  = 1  − 7-s + (−0.866 − 0.5i)11-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s − i·23-s + (0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.5 − 0.866i)37-s + i·41-s − i·43-s + (0.5 + 0.866i)47-s + 49-s − i·53-s + (−0.866 + 0.5i)59-s + 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(251.467\)
Root analytic conductor: \(251.467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (1:\ ),\ -0.992 + 0.118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02060728556 - 0.3466116094i\)
\(L(\frac12)\) \(\approx\) \(0.02060728556 - 0.3466116094i\)
\(L(1)\) \(\approx\) \(0.8158684434 - 0.1020525364i\)
\(L(1)\) \(\approx\) \(0.8158684434 - 0.1020525364i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 - T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + iT \)
43 \( 1 - iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 - T \)
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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

−19.80865270764892812094018883389, −18.86603626199887694008234654916, −18.65913840686463955758506181119, −17.55970582085554710126966012787, −16.95369522862641922849069348411, −16.10239580862574510003525030975, −15.60128285267173903472414117339, −14.86768157767725257665625021316, −13.899605290753789888363399683347, −13.30434633274899247410864108429, −12.48196168080007757371109688532, −12.054093012784296200566671686699, −10.93841849799030178925007291233, −10.12823670211944932936159281925, −9.746680928734027439751151475358, −8.78231471373194494208785853965, −7.91420096427202623233369397490, −7.19443859125390099788951243383, −6.40040508020820959777360758223, −5.5782401627106265899744461890, −4.82222522277967936675477737534, −3.76532351965149692016892443202, −3.034921795846091754813278327942, −2.21037223506954228196097982492, −1.04031944005483333309764173497, 0.07642716705284089733053531145, 0.85234317586180183153770590892, 2.278936801731701274348448920131, 2.92109122187128470478328881120, 3.790391748062476793488839446299, 4.69482341526650521569153795611, 5.679686991177314352318133093955, 6.31778450147276739035444937157, 7.05645827641372342599644051677, 8.10039278274380168712450499051, 8.61117575563338041460433348165, 9.6100780571479450521441615429, 10.33324812655402196160847996589, 10.80557123482517507604998308574, 11.92119371635843954248200599272, 12.631782763182750872130041648299, 13.16263098214792223542206070471, 13.943626707059884467389443889, 14.75221200363810121073464071541, 15.65070842627957234293807309348, 16.07670864195631654878571100414, 16.8931777847028916432403580041, 17.52632669817475153801924899527, 18.54093260096783407819050979990, 19.08411136153536176528755649181

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