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Properties

Label 1-3021-3021.5-r0-0-0
Degree $1$
Conductor $3021$
Sign $-0.934 - 0.354i$
Analytic cond. $14.0294$
Root an. cond. $14.0294$
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  + (−0.265 + 0.964i)2-s + (−0.859 − 0.511i)4-s + (0.291 + 0.956i)5-s + (0.996 − 0.0804i)7-s + (0.721 − 0.692i)8-s + (−0.999 + 0.0268i)10-s + (−0.845 − 0.534i)11-s + (0.653 + 0.757i)13-s + (−0.186 + 0.982i)14-s + (0.476 + 0.879i)16-s + (−0.897 + 0.440i)17-s + (0.239 − 0.970i)20-s + (0.739 − 0.673i)22-s + (−0.984 − 0.173i)23-s + (−0.830 + 0.556i)25-s + (−0.903 + 0.428i)26-s + ⋯
L(s)  = 1  + (−0.265 + 0.964i)2-s + (−0.859 − 0.511i)4-s + (0.291 + 0.956i)5-s + (0.996 − 0.0804i)7-s + (0.721 − 0.692i)8-s + (−0.999 + 0.0268i)10-s + (−0.845 − 0.534i)11-s + (0.653 + 0.757i)13-s + (−0.186 + 0.982i)14-s + (0.476 + 0.879i)16-s + (−0.897 + 0.440i)17-s + (0.239 − 0.970i)20-s + (0.739 − 0.673i)22-s + (−0.984 − 0.173i)23-s + (−0.830 + 0.556i)25-s + (−0.903 + 0.428i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $-0.934 - 0.354i$
Analytic conductor: \(14.0294\)
Root analytic conductor: \(14.0294\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (0:\ ),\ -0.934 - 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1746839608 + 0.9526216176i\)
\(L(\frac12)\) \(\approx\) \(-0.1746839608 + 0.9526216176i\)
\(L(1)\) \(\approx\) \(0.6603383223 + 0.5782102169i\)
\(L(1)\) \(\approx\) \(0.6603383223 + 0.5782102169i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.265 + 0.964i)T \)
5 \( 1 + (0.291 + 0.956i)T \)
7 \( 1 + (0.996 - 0.0804i)T \)
11 \( 1 + (-0.845 - 0.534i)T \)
13 \( 1 + (0.653 + 0.757i)T \)
17 \( 1 + (-0.897 + 0.440i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.379 + 0.925i)T \)
31 \( 1 + (0.534 + 0.845i)T \)
37 \( 1 + (0.748 + 0.663i)T \)
41 \( 1 + (-0.133 - 0.991i)T \)
43 \( 1 + (0.783 + 0.621i)T \)
47 \( 1 + (-0.730 + 0.682i)T \)
59 \( 1 + (-0.545 - 0.837i)T \)
61 \( 1 + (-0.997 + 0.0670i)T \)
67 \( 1 + (-0.488 - 0.872i)T \)
71 \( 1 + (-0.621 + 0.783i)T \)
73 \( 1 + (0.556 - 0.830i)T \)
79 \( 1 + (0.579 + 0.815i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.994 + 0.107i)T \)
97 \( 1 + (0.730 + 0.682i)T \)
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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

−18.38913143998400147965997634315, −18.06764440824319308183748814183, −17.53915589111821226179398630117, −16.83159357490280856045761288859, −15.87564625254734424348294225935, −15.26541138066612850902063989556, −14.19218457760252971150906542883, −13.35094750573013007589487522956, −13.162294183431243891956385446222, −12.157885218732272202079303721925, −11.648153137530027600102622144093, −10.85128901799545279720062289118, −10.15414759309147078880145684039, −9.45981783698227168631992759142, −8.655937250563861352514784693409, −8.03685710681594696996639244819, −7.60098593223523641631219679778, −6.01853437586373526738532062775, −5.31005394192173117389965736421, −4.51146781562545434598622779011, −4.1166543024529993884848578262, −2.72450566900965066272163947974, −2.12345388798376196887108314150, −1.30828442474948596001897165655, −0.33326593317095338864025631744, 1.2918750293839367563517570650, 2.112416136913499954814808563454, 3.23955353282557903961542895455, 4.25937495778469910496223247161, 4.90205128717277010148989423929, 5.90677125221753640548067321289, 6.376641985115423868057963102630, 7.166935457914688133256057598411, 7.97712698105129664242717831424, 8.48478937309041737357495433031, 9.28515628348200428057315215584, 10.290897718727769460142837766873, 10.80193189184436063871172387260, 11.35555274876636220520351947397, 12.552555089024695437232588694355, 13.58525295317111360251462535189, 13.96285658400606811079224857098, 14.50173885573465438838738584030, 15.349398934507596184124416342634, 15.84821119578916202202895925398, 16.62108554998769967339099289377, 17.52253879528264383606207165697, 18.03388277932189322572625563183, 18.42206316247417490155278840161, 19.15290497453430196604338421941

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