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Properties

Label 1-2015-2015.244-r1-0-0
Degree $1$
Conductor $2015$
Sign $-0.465 - 0.885i$
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  + (−0.104 − 0.994i)2-s + (0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + (−0.5 − 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)14-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.809 − 0.587i)18-s + (−0.913 − 0.406i)19-s + (−0.809 + 0.587i)21-s + (0.669 − 0.743i)22-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + (−0.5 − 0.866i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.669 + 0.743i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)14-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (−0.809 − 0.587i)18-s + (−0.913 − 0.406i)19-s + (−0.809 + 0.587i)21-s + (0.669 − 0.743i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.465 - 0.885i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ -0.465 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.136986776 - 1.881966683i\)
\(L(\frac12)\) \(\approx\) \(1.136986776 - 1.881966683i\)
\(L(1)\) \(\approx\) \(0.9670242951 - 0.6516176959i\)
\(L(1)\) \(\approx\) \(0.9670242951 - 0.6516176959i\)

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 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (0.913 + 0.406i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
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

−19.612009803258703951090700496560, −19.260747622811821523246756079, −18.78529907098174316062523523861, −17.64205031954553288628198712862, −16.70466306940070530984669998372, −16.43208273243365807553603443321, −15.57541801534498262216270501204, −14.96399204764769154243032182139, −14.20042785341010391559747540299, −13.66850467083226603900081085992, −12.958673164264437497572078549912, −12.156590651631045674896307515, −10.71936817953965958890781478482, −10.0026046272294666613131848868, −9.47697186546400595367178575159, −8.61609404285572282304189612337, −8.09255579668964967656114960625, −7.26839795692215563431338622915, −6.2555499848961098825792048939, −5.8663179578253820880436752157, −4.51954806613567551946803607891, −3.82227818939174031549387398020, −3.30517290346599696289806916865, −1.95631856666955901905873699119, −0.70827399681470716813669525482, 0.49349804185636550419256383580, 1.49261014659903723833394456548, 2.393414957198268245601558648175, 3.00620663783006394012591346985, 3.882728243542301864425805240477, 4.52550335416192586968090269644, 5.83360430466625403057013496362, 6.81007003628409629882044112141, 7.567723570248652288250249661182, 8.541934906495559646803204492416, 9.194979667398680377807908373853, 9.74893878748960800486068874385, 10.38051396955992495672465146419, 11.58153733231586682269202870966, 12.29784165126049228649310097014, 12.80648569812286707980961830274, 13.41979474611828897598391135200, 14.39954050470943523620185772191, 14.7132528529973558629135784639, 15.89021729223307447000194637431, 16.653896690528952040618438131, 17.75487901558312258207603767372, 18.21487343600301072040567413049, 19.10181774151050770368177378456, 19.50795227334342347352263245369

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