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Properties

Label 1-2015-2015.1012-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.222 - 0.974i$
Analytic cond. $9.35762$
Root an. cond. $9.35762$
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  + (0.104 + 0.994i)2-s + (−0.587 + 0.809i)3-s + (−0.978 + 0.207i)4-s + (−0.866 − 0.5i)6-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.309 − 0.951i)9-s + (0.207 + 0.978i)11-s + (0.406 − 0.913i)12-s + (−0.669 + 0.743i)14-s + (0.913 − 0.406i)16-s + (−0.207 + 0.978i)17-s + (0.913 − 0.406i)18-s + (−0.994 + 0.104i)19-s + (−0.994 + 0.104i)21-s + (−0.951 + 0.309i)22-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.587 + 0.809i)3-s + (−0.978 + 0.207i)4-s + (−0.866 − 0.5i)6-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.309 − 0.951i)9-s + (0.207 + 0.978i)11-s + (0.406 − 0.913i)12-s + (−0.669 + 0.743i)14-s + (0.913 − 0.406i)16-s + (−0.207 + 0.978i)17-s + (0.913 − 0.406i)18-s + (−0.994 + 0.104i)19-s + (−0.994 + 0.104i)21-s + (−0.951 + 0.309i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.222 - 0.974i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1012, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.222 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4923935778 + 0.6177431243i\)
\(L(\frac12)\) \(\approx\) \(-0.4923935778 + 0.6177431243i\)
\(L(1)\) \(\approx\) \(0.4230394031 + 0.6768322462i\)
\(L(1)\) \(\approx\) \(0.4230394031 + 0.6768322462i\)

Euler product

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

−19.36095284291234765174881727561, −18.84519116290338087365808464675, −18.014252983496849268913590441235, −17.40436740420494136845474953537, −16.90517216929006955755630401534, −15.91710938136411554479340188743, −14.64195049870196592330292980396, −13.96257110201565372386647312433, −13.415287679194668394017682682596, −12.84514617261974663502229580833, −11.7609016759240484983137048769, −11.36299966751027956742748323767, −10.85234195109495696011982651264, −9.97520512161161480430641068536, −8.95618673693448261233219599331, −8.14560034017595756979144516195, −7.46664950860812525490723068535, −6.4093451982099131834832181558, −5.58374054692676058277194693077, −4.77156729879397423411987809309, −3.99955200221282494789778687668, −2.91213925592960101083559176556, −2.00966983099167413670529423338, −1.135235148617001696718339334755, −0.32522797759511888972821534019, 1.39054534133267207516741570592, 2.722950740372579914918008305711, 4.02467777720952506969987312863, 4.5227322959697098227831019434, 5.18779770455268829201873307628, 6.09818642079356568936658745207, 6.581888543720866537388720687471, 7.6738519058943168082603233262, 8.63020382383246894550159978095, 9.01775500829136473238247235631, 10.02814220432905227472173745965, 10.66730818239603538534661890598, 11.66491994616502607103849431582, 12.51288129478892410888611185622, 12.94613266527429759517605566644, 14.48841994970572443445951327906, 14.70739376794922370583649945672, 15.28716590929104375125536852261, 16.03652334959775709289301620898, 16.87483283252208290256404685200, 17.33079718286448869055401425315, 18.054052416544853755392834957246, 18.60873701148388177047302169247, 19.74433511506586407195905295282, 20.69124595753659869881294018814

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