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Properties

Label 1-1045-1045.197-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.181 + 0.983i$
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  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s − i·7-s + i·8-s + (0.5 + 0.866i)9-s + i·12-s + (0.866 − 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + i·18-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s − i·7-s + i·8-s + (0.5 + 0.866i)9-s + i·12-s + (0.866 − 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + i·18-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.840947271 + 2.363708822i\)
\(L(\frac12)\) \(\approx\) \(2.840947271 + 2.363708822i\)
\(L(1)\) \(\approx\) \(2.103342677 + 1.066954764i\)
\(L(1)\) \(\approx\) \(2.103342677 + 1.066954764i\)

Euler product

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

−21.27260508388076901050222941358, −20.68849181821281278507917590640, −19.9577022519386742966716666476, −19.01063568247674841610709060821, −18.63967048270924603321413705270, −17.88663079860648512368957285790, −16.19587755471501499041649436370, −15.78125276961469236642066421545, −14.646391649306893438768348206992, −14.36692726317733875879643840460, −13.42887693715537747689651451410, −12.70479526449313989971840636250, −12.01642715580844907346231908173, −11.33576013475675737081400714450, −10.11807123002802451368579438019, −9.32832381138973525836089264226, −8.531547147717808509992475526, −7.51916663368226761705114323557, −6.4471168163510044095694305232, −5.84443052466280492227182588429, −4.70053906809813750830117639981, −3.67286417902271068111612827835, −2.90817874216160681656346659885, −2.077751159895397055075396450578, −1.18879437409088069633106130123, 1.48442925018789779043808544459, 2.72288689973096867528365850678, 3.72934099995907654755180703383, 4.0193840952287730162757848192, 5.1644219286617720618331333417, 6.086012754269208724173689960, 7.15267081881795577812048882758, 7.97631935778396301274538211858, 8.423462668667290588872955829368, 9.78552919339109253516940597754, 10.480034879072187279865495306414, 11.39324528677532962924858211582, 12.47858806852871328246833771221, 13.48955755208176679281761103534, 13.731916960651810600887009428059, 14.58861308464492264036201116206, 15.40819691856506190769791855779, 15.96719310177040430630824667544, 16.83576833224540817628157449579, 17.48127252912359205117140042532, 18.75338203702539603517240697557, 19.6600163873599083831922790329, 20.61444454601138339937693033771, 20.71698668662917629986906284748, 21.73273868207310752500017278812

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