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Properties

Label 1-2015-2015.228-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.184 + 0.982i$
Analytic cond. $216.541$
Root an. cond. $216.541$
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  + (0.913 + 0.406i)2-s + (−0.406 − 0.913i)3-s + (0.669 + 0.743i)4-s − i·6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.669 + 0.743i)9-s + (0.951 + 0.309i)11-s + (0.406 − 0.913i)12-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.951 − 0.309i)17-s + (−0.913 + 0.406i)18-s + (0.587 + 0.809i)19-s + (0.994 − 0.104i)21-s + (0.743 + 0.669i)22-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.406 − 0.913i)3-s + (0.669 + 0.743i)4-s − i·6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.669 + 0.743i)9-s + (0.951 + 0.309i)11-s + (0.406 − 0.913i)12-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.951 − 0.309i)17-s + (−0.913 + 0.406i)18-s + (0.587 + 0.809i)19-s + (0.994 − 0.104i)21-s + (0.743 + 0.669i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.184 + 0.982i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.184 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.506542466 + 3.019968157i\)
\(L(\frac12)\) \(\approx\) \(2.506542466 + 3.019968157i\)
\(L(1)\) \(\approx\) \(1.678622595 + 0.5611218768i\)
\(L(1)\) \(\approx\) \(1.678622595 + 0.5611218768i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.207 + 0.978i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (0.743 + 0.669i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.207 + 0.978i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−19.723047726934494256736373904289, −19.25075233214637776392544636776, −18.03049571731090368491770383326, −17.02914974790622555601004587669, −16.5649801447775400889551626538, −15.90275117874406589283845034428, −15.02881541459290198518573613790, −14.38771455768348371997451848563, −13.81134148420672408259630282847, −12.92236732066297597544225008079, −12.05293251932876595192677980392, −11.41643132340892246219286701415, −10.75801099060794940619206942996, −10.027296427597792742216362243662, −9.503039572092940256228194239813, −8.418769816173757844032620400758, −7.04235371267326441375009169093, −6.55157833261803881558984586551, −5.63189528565163354823303954254, −4.85723332101246068320090148117, −4.097501093373335413307301583884, −3.49375608510421060180376110797, −2.76260731840372653140368409035, −1.20120309497694862419254315516, −0.56985242983352498683576656560, 1.13754077262511554653249244651, 1.97417743220574731781203514135, 2.97431587761644605133423453413, 3.66067403919983632930975640064, 5.07070941006142453802509266793, 5.43115821683059858509395956840, 6.35304972105293122363762497693, 6.86507222562590764883192000618, 7.70302427766796336790065477536, 8.46781276142788898818336610836, 9.376186927285080453877246306142, 10.534186070096838664866030073284, 11.63412086760696805750954802098, 12.06111213768592785422631168445, 12.42893513760601077249452929174, 13.37178119180038986560215224547, 14.09139641313480926152900730390, 14.64605185960217668910029428683, 15.58622751113167161884020055278, 16.24485950063009575742738665415, 17.10545091510524912776139556776, 17.51838142850351716283092366644, 18.668653810450902507790302918933, 19.029639811878339389081467738590, 20.05093639591485890680164712803

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