Properties

 Label 1-1045-1045.987-r1-0-0 Degree $1$ Conductor $1045$ Sign $-0.952 - 0.304i$ Analytic cond. $112.300$ Root an. cond. $112.300$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Learn more

Dirichlet series

 L(s)  = 1 + (0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s − i·12-s + (−0.951 − 0.309i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.951 − 0.309i)17-s + (−0.587 + 0.809i)18-s + 21-s − i·23-s + (0.309 − 0.951i)24-s + ⋯
 L(s)  = 1 + (0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s − i·12-s + (−0.951 − 0.309i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.951 − 0.309i)17-s + (−0.587 + 0.809i)18-s + 21-s − i·23-s + (0.309 − 0.951i)24-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $-0.952 - 0.304i$ Analytic conductor: $$112.300$$ Root analytic conductor: $$112.300$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (987, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1045,\ (1:\ ),\ -0.952 - 0.304i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.03084677653 + 0.1980978275i$$ $$L(\frac12)$$ $$\approx$$ $$0.03084677653 + 0.1980978275i$$ $$L(1)$$ $$\approx$$ $$1.203945798 + 0.1598498180i$$ $$L(1)$$ $$\approx$$ $$1.203945798 + 0.1598498180i$$

Euler product

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

−21.13431879898878042750280064076, −20.34094290329756071362383978308, −19.590342513420117088989756525132, −18.8996715886785376500231901849, −17.44490411275006227339119982110, −16.86147104429803012207149703240, −16.12990856039856542988512441677, −15.39572450562564005488933723613, −14.585452279308258991887458139414, −13.873623810128858771002505407922, −12.93833284835572385713709526211, −12.06926177111818730759886317366, −11.530744604247196155249247013671, −10.395194289354174545807220383172, −10.09953488334600399914866535629, −9.22915680740695451735426956149, −7.58030450798344755037795542506, −6.83528119987516026367347231521, −5.86069840546595857065221241175, −5.19730682338347870979082782124, −4.16868060465220454421544118435, −3.6672626696515505555794986398, −2.64561553574596628414434294037, −1.20506465470354380024443396180, −0.03089126290508591846054401165, 1.51667937740179843593443178978, 2.6275761063593235402595271780, 3.23821687671132911390427839965, 4.89248389282864512028217465094, 5.24109971472150062402697435396, 6.330839114278784880465933941726, 6.79201070929831621298239583693, 7.77982607027254287860747258581, 8.54692205543585943484196657436, 9.944937950205059560207110653098, 10.84155268630716874871981898561, 12.0336698722228555988061965339, 12.21123886396762628613958768283, 12.91487654151980437628328773926, 13.84703422024241514041573921634, 14.598432253094497986933304710419, 15.436257171248525673403651708811, 16.43308397194014823295129727670, 16.78810864589233407055504470421, 17.882825417532113813034776471581, 18.596515256074176955657360750727, 19.53271018521185510164267264773, 20.13899660580298144150816052138, 21.41396889525858414828064360294, 21.89224778664060598577471677835