↑

Properties

Label 1-2015-2015.57-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.902 - 0.430i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 − 0.5i)6-s + (0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + 16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 − 0.5i)6-s + (0.5 + 0.866i)7-s + 8-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 + 0.866i)14-s + 16-s + (0.866 − 0.5i)17-s + (0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s + (0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.902 - 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(9.072637589 - 2.050595953i\)
\(L(\frac12)\) \(\approx\) \(9.072637589 - 2.050595953i\)
\(L(1)\) \(\approx\) \(3.318515418 - 0.4291659484i\)
\(L(1)\) \(\approx\) \(3.318515418 - 0.4291659484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
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 \( 1 + iT \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.929246290148147760794226066695, −19.5038778376312411240117108089, −18.61881705267038083148142645129, −17.27105799611805379413830313987, −16.73695672016339377766618281946, −15.980792136141388358005880609413, −15.2654924182749084527469775192, −14.44620258382145718309834029364, −13.97428328248451015822525968892, −13.61877418778859463141591420123, −12.54054237354324777261976511154, −11.72948027777372573382594952248, −10.96947872168426345437502562127, −10.209971568851367927924365399584, −9.53933082456103301039776258134, −8.321476329074753719767020937140, −7.76207165629033147106990820061, −6.99718185573328697986621091811, −6.00068093154423645593807424715, −5.05354276712438822691461750614, −4.35630878197004028739100091836, −3.43522188742516738717616122190, −3.23729718302029803498137535657, −1.73520303393874812314595549009, −1.218549329999223683480586718872, 1.04894096451542021816477514719, 1.80116637691260646142325320965, 2.71691328024423345836430525345, 3.26402771663509263385144784650, 4.37123096713819569086449438471, 5.03152072281135144250189470946, 6.08594885611839244039901557044, 6.76938150229113980996294149765, 7.57311764580794845904922880707, 8.27879772615355161615175410575, 9.21736091058637020768426922324, 9.923607586531935852414625354895, 11.12884904770516554114461130214, 12.009985763116697522177615769, 12.26895184371581802156320871815, 13.11139725007165457312254733344, 14.106916745831243410092278620769, 14.38412729944621572120378078555, 15.049611650170864587269855421502, 15.78720213420641987189235396180, 16.54878859727115397149194743767, 17.712555611159175726464027800275, 18.28491438653218990428803176213, 19.24251954178006018131039761415, 19.78876835619834116337074856638

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