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Properties

Label 1-3015-3015.1757-r0-0-0
Degree $1$
Conductor $3015$
Sign $-0.997 + 0.0641i$
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  + (0.0950 + 0.995i)2-s + (−0.981 + 0.189i)4-s + (−0.0950 − 0.995i)7-s + (−0.281 − 0.959i)8-s + (−0.928 + 0.371i)11-s + (−0.690 − 0.723i)13-s + (0.981 − 0.189i)14-s + (0.928 − 0.371i)16-s + (0.755 + 0.654i)17-s + (−0.415 − 0.909i)19-s + (−0.458 − 0.888i)22-s + (−0.998 + 0.0475i)23-s + (0.654 − 0.755i)26-s + (0.281 + 0.959i)28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.0950 + 0.995i)2-s + (−0.981 + 0.189i)4-s + (−0.0950 − 0.995i)7-s + (−0.281 − 0.959i)8-s + (−0.928 + 0.371i)11-s + (−0.690 − 0.723i)13-s + (0.981 − 0.189i)14-s + (0.928 − 0.371i)16-s + (0.755 + 0.654i)17-s + (−0.415 − 0.909i)19-s + (−0.458 − 0.888i)22-s + (−0.998 + 0.0475i)23-s + (0.654 − 0.755i)26-s + (0.281 + 0.959i)28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3015\)    =    \(3^{2} \cdot 5 \cdot 67\)
Sign: $-0.997 + 0.0641i$
Analytic conductor: \(14.0016\)
Root analytic conductor: \(14.0016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3015} (1757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3015,\ (0:\ ),\ -0.997 + 0.0641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009601837254 + 0.2991714432i\)
\(L(\frac12)\) \(\approx\) \(0.009601837254 + 0.2991714432i\)
\(L(1)\) \(\approx\) \(0.7060898298 + 0.2572623166i\)
\(L(1)\) \(\approx\) \(0.7060898298 + 0.2572623166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.0950 + 0.995i)T \)
7 \( 1 + (-0.0950 - 0.995i)T \)
11 \( 1 + (-0.928 + 0.371i)T \)
13 \( 1 + (-0.690 - 0.723i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-0.998 + 0.0475i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.327 + 0.945i)T \)
43 \( 1 + (0.189 - 0.981i)T \)
47 \( 1 + (-0.458 + 0.888i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (0.371 + 0.928i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
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

−18.69255062636910059416155247410, −18.33111898287125925584125085284, −17.63275389357746165269933757014, −16.5120293039133075488867382272, −16.07842903543253551732016221720, −14.98006164327527956446316646278, −14.37353628543598568149991527636, −13.78840772480895505888246456177, −12.8193354325525016465379944921, −12.22310042446974737670559699174, −11.868968584256298126759214481780, −10.880921019285341817277824684883, −10.24551228802182164781180939433, −9.51904474285622815859827763618, −8.85974239531997045220858151894, −8.13550424811222516096872075441, −7.31690746344106769494552198269, −6.0209120286547714789292270695, −5.45488396485713265284727826128, −4.77606187838848657170577584147, −3.782677657323483509034115784969, −2.96693459826948832316007105266, −2.28060180934071783625016694205, −1.55607824694556387600401792174, −0.10664170354094349848887737595, 0.883058060726320383301696695335, 2.306914308227565394902114778749, 3.32154261390469517575292669512, 4.18933221032738337888642766140, 4.81013110147975928058067015311, 5.64262869208053725933060225645, 6.37405228952572542474763901392, 7.25131203566046714630835136809, 7.86650105789544481157796741525, 8.21598019768155988215016335940, 9.523656105656296245211088644281, 10.01994224202440521973102589560, 10.60229516545488367735580902217, 11.74342317443221041818653202425, 12.717722156929689799194861868662, 13.18382098347313169471318468515, 13.76640096624134480203162590132, 14.72224904978428522052020614096, 15.142362875842083872900511484294, 15.89889773210759525843502987187, 16.62711037786794698092276716321, 17.35887890336095192100809437517, 17.62367055500019755464909576365, 18.59569345256473754498715610338, 19.26304781191560612510637451275

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