Properties

Label 1-1045-1045.604-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.683 - 0.730i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)12-s + (0.939 + 0.342i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s − 18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.173 + 0.984i)24-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)12-s + (0.939 + 0.342i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s − 18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (−0.173 + 0.984i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.683 - 0.730i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.683 - 0.730i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6394048769 - 0.2773500525i\)
\(L(\frac12)\) \(\approx\) \(0.6394048769 - 0.2773500525i\)
\(L(1)\) \(\approx\) \(0.5741221942 - 0.1409435647i\)
\(L(1)\) \(\approx\) \(0.5741221942 - 0.1409435647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−21.76994428400527181479188003888, −20.88079869881536551726592692548, −19.724233325663012457644619070156, −19.032842534760985091259662427238, −18.22282999455384945627145627520, −17.99875261718715104330335187487, −16.90499152139931709774034193104, −16.217926673550906416602458568761, −15.71377231449743085093952149171, −14.883700305801028268979271514886, −13.702777580928851658832259467832, −12.97209855218660498615176163700, −11.82500678257117563841102193446, −11.371159448864525116084947474572, −10.25408349344611390237007620777, −9.67956287344298447567902126726, −8.65172959755160746667988761024, −7.81842017381187829657917132161, −6.94035892472365063345260638219, −6.04587205033114715671665965929, −5.644114010975752425999873479304, −4.68509769482985860314344702936, −3.10239049397874004890322494389, −1.80291838675969911208932223396, −0.79609891343128066718299472790, 0.68716957023608528250070063717, 1.53191875475091564125880533160, 3.08976516651447014826756682579, 3.92280941989511793609440902548, 4.6486936091020674389222272364, 6.16922769021974934434978168785, 6.66990818980934429924193992368, 7.745823116832812717445684127008, 8.6650060198417770666853652618, 9.72612269035481920157392159194, 10.319366496069179231828837531187, 10.8724091814830793526981907160, 11.73911895403181668547840238197, 12.50764611991520267205197562152, 13.221644031295503956130009449939, 14.23694071610488916457780560637, 15.66646182844677558215365384117, 16.177526635598374224475649596730, 16.9730962991715682540038817107, 17.37349526100911961305442475146, 18.45429656220504321237891683027, 18.89621270582588427131653136288, 19.96043806401467551118862627936, 20.63258840035442512454686784365, 21.411405869687344770496559198783

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