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Properties

Label 1-3015-3015.263-r0-0-0
Degree $1$
Conductor $3015$
Sign $-0.659 + 0.751i$
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  + (−0.618 − 0.786i)2-s + (−0.235 + 0.971i)4-s + (0.618 + 0.786i)7-s + (0.909 − 0.415i)8-s + (0.888 + 0.458i)11-s + (−0.814 + 0.580i)13-s + (0.235 − 0.971i)14-s + (−0.888 − 0.458i)16-s + (0.281 − 0.959i)17-s + (0.142 − 0.989i)19-s + (−0.189 − 0.981i)22-s + (−0.945 + 0.327i)23-s + (0.959 + 0.281i)26-s + (−0.909 + 0.415i)28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (−0.618 − 0.786i)2-s + (−0.235 + 0.971i)4-s + (0.618 + 0.786i)7-s + (0.909 − 0.415i)8-s + (0.888 + 0.458i)11-s + (−0.814 + 0.580i)13-s + (0.235 − 0.971i)14-s + (−0.888 − 0.458i)16-s + (0.281 − 0.959i)17-s + (0.142 − 0.989i)19-s + (−0.189 − 0.981i)22-s + (−0.945 + 0.327i)23-s + (0.959 + 0.281i)26-s + (−0.909 + 0.415i)28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3015\)    =    \(3^{2} \cdot 5 \cdot 67\)
Sign: $-0.659 + 0.751i$
Analytic conductor: \(14.0016\)
Root analytic conductor: \(14.0016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3015} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3015,\ (0:\ ),\ -0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1102039202 + 0.2433192239i\)
\(L(\frac12)\) \(\approx\) \(0.1102039202 + 0.2433192239i\)
\(L(1)\) \(\approx\) \(0.6799065203 - 0.1030416028i\)
\(L(1)\) \(\approx\) \(0.6799065203 - 0.1030416028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.618 - 0.786i)T \)
7 \( 1 + (0.618 + 0.786i)T \)
11 \( 1 + (0.888 + 0.458i)T \)
13 \( 1 + (-0.814 + 0.580i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (-0.945 + 0.327i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.723 + 0.690i)T \)
43 \( 1 + (-0.971 + 0.235i)T \)
47 \( 1 + (-0.189 + 0.981i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (-0.458 + 0.888i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
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

−18.63468412274095121029655588282, −18.011093179238763309727923078064, −17.27745947272532267802799488502, −16.64054225871373495308622471564, −16.444944404942088798554388327366, −15.06181250493174559072702889826, −14.80323218241310330158702596100, −14.10462201488819975048743027081, −13.471848545494832307889779480076, −12.40912658251431070881963689436, −11.638405504941615205937751685804, −10.66159667144999881417155159237, −10.25093813051839303876217297591, −9.524061418344034187856830178056, −8.4451536010341192687473147095, −8.14586633045643552809980926785, −7.25944873339109994263588588579, −6.65889005698509869003344889313, −5.72673592307169318872693108088, −5.1388118474991501850994491424, −4.12156552603165411228679176229, −3.46679526018428887078886059729, −1.84359496823743367462402362108, −1.37900195918216233591002946386, −0.09970357005358519817136994375, 1.31235278818921096677773434381, 2.07777847861289565923590951512, 2.68156341112724601011432560297, 3.75462921428906373767630104453, 4.5695181918320077582168486622, 5.21825855090405660727029352682, 6.419861525805798332301899256028, 7.3162839774325612050596409641, 7.84511623349739754882084304097, 8.83281046766111457145968522431, 9.49122108399666667463071009715, 9.72206332177879774070307812708, 11.04218760535967741218196264203, 11.508616295302083694269931487762, 12.06925435015466544214641171414, 12.64183028656163065770053805461, 13.70337385692822332966832528253, 14.29777222387131159977838233516, 15.12815122671879798753210485191, 15.93351342169078108168653875634, 16.79783332813973073368454882096, 17.32833519517240271818039781136, 18.09865746351675238345058752200, 18.50902433670373692720423908193, 19.390964033503034902835306204189

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