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Properties

Label 1-1045-1045.949-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.999 + 0.0237i$
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  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s − 21-s − 23-s + (0.309 − 0.951i)24-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (0.309 − 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (−0.809 − 0.587i)18-s − 21-s − 23-s + (0.309 − 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.999 + 0.0237i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009467320189 - 0.7964083104i\)
\(L(\frac12)\) \(\approx\) \(0.009467320189 - 0.7964083104i\)
\(L(1)\) \(\approx\) \(0.7778599419 - 0.3613306766i\)
\(L(1)\) \(\approx\) \(0.7778599419 - 0.3613306766i\)

Euler product

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

−21.82768826426818579878033154169, −21.36862924932134732964904452059, −20.21960764167631863329727470543, −19.05858162446041556568196231363, −18.423004778511290925253493881096, −17.51616632397512488759444077480, −17.17510096425681770411532536520, −16.37088874179692718610621061752, −15.6049874234491584635810168818, −14.56316779099436160864567724024, −13.8371279233615946123460460086, −13.26446837024932747357274008556, −12.27140514539885830608940279100, −11.578998592894894044213018221181, −10.69803065020358078455853096897, −9.66240532068794979158842911223, −8.35329167447794720577394737629, −7.9141906201865115714098972591, −6.90195871743422160288312652419, −6.32008756448897472331870021621, −5.46183486269126936861825498704, −4.473799470451248177794155002699, −3.94585286156627774940328551506, −2.15621704318440850872886505098, −1.06324520053617915860879639500, 0.197051026998441583935793052592, 1.223842159465888718921597642070, 2.398422865043634486564500276480, 3.44922046065494939072802427345, 4.375345453656483989279768838342, 5.30082711763764802326233633215, 5.60936053872307226180709252419, 6.899081462615587498084311213443, 8.34686408302297304385327061658, 9.021663429168838148674674719426, 10.056568405167752311034310826284, 10.645196043926126303753501262464, 11.42072388197415946336633494810, 12.0620464190589867346182097966, 12.69118841451825986247808431309, 13.843715164911281342286331028306, 14.58521085631505496645916673152, 15.47086588110059533476398322630, 16.083642887214050026874574915297, 17.38619060225986837275396679263, 18.08045980369217966412657970718, 18.28141013512749593934714133374, 19.63545911209379114243358318184, 20.393638049113218901633598595522, 21.06593982294968138082152426980

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