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Properties

Label 1-2015-2015.37-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.308 - 0.951i$
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  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s + 7-s − 8-s + (0.5 − 0.866i)9-s + i·11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + i·17-s + (−0.5 − 0.866i)18-s − i·19-s + (0.866 − 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s − i·6-s + 7-s − 8-s + (0.5 − 0.866i)9-s + i·11-s + (−0.866 − 0.5i)12-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + i·17-s + (−0.5 − 0.866i)18-s − i·19-s + (0.866 − 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.880613301 - 2.821274930i\)
\(L(\frac12)\) \(\approx\) \(3.880613301 - 2.821274930i\)
\(L(1)\) \(\approx\) \(1.690510820 - 1.084919067i\)
\(L(1)\) \(\approx\) \(1.690510820 - 1.084919067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + T \)
11 \( 1 + iT \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + iT \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + iT \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−20.20825444699033471263277865112, −18.87043373525333796513744597559, −18.61211235221688318839137101388, −17.52248670521710504855692973832, −16.75786806180546216276873255532, −16.12409883190710485282703615949, −15.48391015053008826565667382730, −14.563541615815709715264953948716, −14.32104703000706996345731123561, −13.58724365245354287023771526984, −12.88635336699969199803173887444, −11.797738504736830210674169710, −11.09474662086477677689825381790, −10.11615259477399515289784750234, −9.001730196012355334045866569988, −8.70222357203770522732172860967, −7.71125056270805996514743187846, −7.41595399845411996737029011347, −6.09079456538675625253222227978, −5.33934877595089143915848247979, −4.5733841146036888207404828737, −3.85585177576793013500014318458, −3.0170588335290089013317883605, −2.14579032326667867018789684967, −0.69530810607707936683308630388, 0.94334931660651538192804251735, 1.64307037884539002141029892108, 2.33397346730958570825223668272, 3.19695545980431726096905154269, 4.168533470687922695439528932706, 4.768180829557648057008975276094, 5.732575219574921004596103365863, 6.871180183736812422130314792866, 7.53543772172231615313130730088, 8.61839397185170265637237400655, 9.06188299122455881449769687340, 10.02542241851349021825054073621, 10.76481084282055774032430392917, 11.64537009188173806273506973205, 12.27496682289633393830687230034, 13.126497795806474240007703105735, 13.50715525975456722366825318240, 14.504415197247463419716701870643, 15.05057190557390445126799133782, 15.312972877694392334591744504014, 16.955674530926593307360198665863, 17.86876371611550190705574558103, 18.15641140083745280349168403324, 19.16026393192397630357865815625, 19.69095781836727159778717175922

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