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Properties

Label 1-1045-1045.18-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 $0$

Origins

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Normalization:  

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: $\chi_{1045} (18, \cdot )$
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 \( 1 + (-0.951 + 0.309i)T \)
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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

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

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