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Properties

Label 1-3015-3015.1622-r0-0-0
Degree $1$
Conductor $3015$
Sign $0.890 - 0.454i$
Analytic cond. $14.0016$
Root an. cond. $14.0016$
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  + (0.189 − 0.981i)2-s + (−0.928 − 0.371i)4-s + (−0.189 + 0.981i)7-s + (−0.540 + 0.841i)8-s + (−0.723 − 0.690i)11-s + (0.998 − 0.0475i)13-s + (0.928 + 0.371i)14-s + (0.723 + 0.690i)16-s + (0.989 + 0.142i)17-s + (0.654 + 0.755i)19-s + (−0.814 + 0.580i)22-s + (0.0950 − 0.995i)23-s + (0.142 − 0.989i)26-s + (0.540 − 0.841i)28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.189 − 0.981i)2-s + (−0.928 − 0.371i)4-s + (−0.189 + 0.981i)7-s + (−0.540 + 0.841i)8-s + (−0.723 − 0.690i)11-s + (0.998 − 0.0475i)13-s + (0.928 + 0.371i)14-s + (0.723 + 0.690i)16-s + (0.989 + 0.142i)17-s + (0.654 + 0.755i)19-s + (−0.814 + 0.580i)22-s + (0.0950 − 0.995i)23-s + (0.142 − 0.989i)26-s + (0.540 − 0.841i)28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3015\)    =    \(3^{2} \cdot 5 \cdot 67\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(14.0016\)
Root analytic conductor: \(14.0016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3015} (1622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3015,\ (0:\ ),\ 0.890 - 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.452752020 - 0.3491698538i\)
\(L(\frac12)\) \(\approx\) \(1.452752020 - 0.3491698538i\)
\(L(1)\) \(\approx\) \(0.9806191505 - 0.3755577292i\)
\(L(1)\) \(\approx\) \(0.9806191505 - 0.3755577292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.189 - 0.981i)T \)
7 \( 1 + (-0.189 + 0.981i)T \)
11 \( 1 + (-0.723 - 0.690i)T \)
13 \( 1 + (0.998 - 0.0475i)T \)
17 \( 1 + (0.989 + 0.142i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.0950 - 0.995i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.786 + 0.618i)T \)
43 \( 1 + (-0.371 - 0.928i)T \)
47 \( 1 + (-0.814 - 0.580i)T \)
53 \( 1 + (0.989 - 0.142i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.281 + 0.959i)T \)
79 \( 1 + (0.888 + 0.458i)T \)
83 \( 1 + (-0.690 + 0.723i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−18.93859609602173663519421230121, −18.04910433250870417739991658642, −17.80589868315674286532320455497, −16.85300262623845492258298465327, −16.22291667848026417717798088968, −15.78888456923284187212789092870, −14.91069142315645042413093477665, −14.2520478054094175666612354074, −13.50644148428749769031041372764, −13.0636253040975614388689872070, −12.340856325920004388654009283567, −11.21889780473779257424626561159, −10.545633668568319013331693117248, −9.553658102246778935269085128967, −9.20547644203515916105818866315, −7.96689503512454242063588702223, −7.52871588954375196116585921886, −6.9857668547724107194812202249, −6.01048672659886646531818838497, −5.30981710292625748155489809621, −4.59425102305480734135830140856, −3.63971582790038827122722548469, −3.18240466231201970417829416142, −1.63425341723942778766631211894, −0.57827816523522150900009017674, 0.84378124631016680249857434528, 1.7683840191310696745274022604, 2.71794963200734547844048482865, 3.3106142177016046173873320469, 4.04822848229680215452874297095, 5.27604946139085375354939200995, 5.62252481943872983651649700089, 6.36798138543419091124330472815, 7.80193879704197194889027563181, 8.44701066701573329224882765875, 8.99534032308158407630125468645, 9.97741369769782590571887484498, 10.42638195849447223064856120359, 11.39057647556603986786640815419, 11.825372272121119122336850460430, 12.68662897076662052214074757158, 13.16787934207067056209487145671, 13.97268415125447828499475711746, 14.66107903799736586170568842483, 15.414469729420184420461365776537, 16.2433598502030427613209469168, 16.870767067490139824460942438172, 18.11633843551038158990964308678, 18.54547152814033073838774316903, 18.76076025889072346388459605070

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