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Properties

Label 1-2015-2015.584-r1-0-0
Degree $1$
Conductor $2015$
Sign $0.275 - 0.961i$
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  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)22-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(216.541\)
Root analytic conductor: \(216.541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (1:\ ),\ 0.275 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.649744381 - 1.997976033i\)
\(L(\frac12)\) \(\approx\) \(2.649744381 - 1.997976033i\)
\(L(1)\) \(\approx\) \(1.532236792 - 0.5079585562i\)
\(L(1)\) \(\approx\) \(1.532236792 - 0.5079585562i\)

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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 - T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−20.05201853433967917505022994281, −19.388184139815951926288442227008, −18.500419064233313249799554169844, −17.5328872305504122550248481508, −16.64334955489171138468817831336, −16.130787815079927242934930988869, −15.40343457001709066648261562910, −15.026983357720330200814714437951, −14.14619802933216517323098426120, −13.02566186664161624442677205658, −12.79228409307852079063165772808, −11.61392383889703641454694146816, −11.19773232219680115479271844097, −10.55535772844608223899060045644, −9.50957154255805271739441429708, −8.84913934330142404431708236093, −7.789952869136253931704062516410, −6.59629368271370978552176693231, −6.04963555751750986299422682773, −5.33777484430169069533365392122, −4.73398408635855611526328672248, −3.663568287285186997015192226267, −3.08880076733765603205674209675, −2.23714561877503371987671907068, −0.73723890677012233504494971616, 0.55894084062902114433507668574, 1.55429848150799107523953062258, 2.45734809906313548763993864064, 3.31849981353173280756856734552, 4.37411725187937188494917876760, 5.09935213131741358865908734044, 5.888845459477238878679774424205, 6.75246163506265738285817442118, 7.367071210488746710521803291396, 7.73770756503056528429350502071, 9.220980672401351965663447156343, 10.32607623885373534274451894381, 10.81824082698562195948389733160, 11.74415902049889452845363731420, 12.335091812784565549433958972066, 13.10773829075788590293518398780, 13.523879058637931968005361590415, 14.22327874107706015044650423703, 15.16864024118726385946221994397, 15.94398162202680372115719882516, 16.758691415585857989227847161847, 17.17808486731422222759576242172, 18.23683199868983561531618759756, 18.89747350846179984044531468536, 19.78081008564757005476942871402

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