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Properties

Label 1-1045-1045.238-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.895 - 0.444i$
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  + (−0.788 + 0.615i)2-s + (0.694 + 0.719i)3-s + (0.241 − 0.970i)4-s + (−0.990 − 0.139i)6-s + (−0.406 + 0.913i)7-s + (0.406 + 0.913i)8-s + (−0.0348 + 0.999i)9-s + (0.866 − 0.5i)12-s + (−0.469 − 0.882i)13-s + (−0.241 − 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.999 + 0.0348i)17-s + (−0.587 − 0.809i)18-s + (−0.939 + 0.342i)21-s + (0.642 + 0.766i)23-s + (−0.374 + 0.927i)24-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)2-s + (0.694 + 0.719i)3-s + (0.241 − 0.970i)4-s + (−0.990 − 0.139i)6-s + (−0.406 + 0.913i)7-s + (0.406 + 0.913i)8-s + (−0.0348 + 0.999i)9-s + (0.866 − 0.5i)12-s + (−0.469 − 0.882i)13-s + (−0.241 − 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.999 + 0.0348i)17-s + (−0.587 − 0.809i)18-s + (−0.939 + 0.342i)21-s + (0.642 + 0.766i)23-s + (−0.374 + 0.927i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4712113352 - 0.1104090888i\)
\(L(\frac12)\) \(\approx\) \(0.4712113352 - 0.1104090888i\)
\(L(1)\) \(\approx\) \(0.6221289955 + 0.3792719457i\)
\(L(1)\) \(\approx\) \(0.6221289955 + 0.3792719457i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.788 + 0.615i)T \)
3 \( 1 + (0.694 + 0.719i)T \)
7 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (-0.469 - 0.882i)T \)
17 \( 1 + (-0.999 + 0.0348i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.961 + 0.275i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.829 - 0.559i)T \)
53 \( 1 + (0.529 - 0.848i)T \)
59 \( 1 + (0.559 - 0.829i)T \)
61 \( 1 + (0.374 + 0.927i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.848 + 0.529i)T \)
73 \( 1 + (0.898 - 0.438i)T \)
79 \( 1 + (-0.990 + 0.139i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.788 - 0.615i)T \)
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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

−21.019556806341850978226385302114, −20.33126856462831539810931843271, −19.90523233749483420677968596165, −18.888291480025704737969127744084, −18.75277869652590868837395983640, −17.4876791636709489965323654848, −17.05606444676900460650602147916, −16.14519419691034696011132322332, −15.09826886401311285470623071494, −14.08599120355025534688893859930, −13.29585413344926137737013740726, −12.7748793765573292494248043720, −11.80918872421330953268138493665, −11.02148621990686254660601586743, −10.0728881710053681223247788179, −9.22071621512381733081026486084, −8.69716565376959398169549521117, −7.54157263189671332617048442059, −7.10082021973879334703519454590, −6.31750002333123610354727746460, −4.431168471885991878565926132, −3.71294085638912349704045571607, −2.66176109235109412963761074596, −1.89711842944990524454591298363, −0.83869938835620342049073050226, 0.13933431986548119423873187297, 1.84304002274598283488579247479, 2.64782625574946431445286747944, 3.72023731694287003501920505007, 5.160428747637798049964573839236, 5.49750055001910136965515872376, 6.81608570716684409536295215860, 7.6137111528796737695326162346, 8.63742071486961968213554138612, 9.041436276010344829070048088222, 9.827330069941854673548681595069, 10.62240781875965048974064218075, 11.41292130795580357291457314835, 12.71509951793430632703066102556, 13.57967449127949867637489028207, 14.62739177451602186088720254799, 15.21460512713579035952893407090, 15.68442030341010250770039323616, 16.459170194299523501321565132756, 17.36298578327125572073567614240, 18.15686884649739205552867118458, 19.01915123710428272204103476709, 19.68715253472151576645459696456, 20.24182860532439371623003411897, 21.21064776643864491758169645054

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