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Properties

Label 1-1045-1045.392-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.339 - 0.940i$
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  + (0.207 − 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (−0.743 − 0.669i)13-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (0.587 + 0.809i)18-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (0.406 + 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s − i·12-s + (−0.743 − 0.669i)13-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (0.587 + 0.809i)18-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.339 - 0.940i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.339 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8039032500 - 1.145438281i\)
\(L(\frac12)\) \(\approx\) \(0.8039032500 - 1.145438281i\)
\(L(1)\) \(\approx\) \(1.069605821 - 0.2419033885i\)
\(L(1)\) \(\approx\) \(1.069605821 - 0.2419033885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.207 - 0.978i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (0.743 + 0.669i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−21.61660125040034579987783266566, −20.8991642700911619112624149651, −19.80991454066970247957962633743, −19.24107007011954238432836279597, −18.11559752650316527813171904717, −17.8327764219154485195753248879, −16.73581947335544503824815203508, −16.42175473792150718381863145533, −14.84466052727023508605888186867, −14.574003267642458351853590295008, −13.90668379227086463669568192391, −13.03695517683539982072110829982, −12.37603102700936669117891652015, −11.47370487309868758968860041399, −10.19188157658729722967356546187, −9.24975708686224323414467323624, −8.31475084522237929498646695288, −7.68556315865774139255293674678, −7.05814577238633137876603242023, −6.24930467902829677218883965117, −5.26957155178628680825482759600, −4.228598206515491800952640949342, −3.40928600003603800028571737026, −2.03272494262965805167029529257, −0.947307339921074105234035357317, 0.2851990932523439427775987188, 1.86428656762481656657919832923, 2.6264753456354674286525136169, 3.462769727836205402984464466486, 4.42983586553080667377446339309, 5.2781669126935527268408254317, 5.77306981627169968404410142406, 7.737433799743288810226864587627, 8.31749025748920235082947020590, 9.481327468248740672134056386116, 9.71120992456919561555362937825, 10.74601342735107434301110421829, 11.55708394288347824981341761902, 12.1310404349868127430219259145, 13.21231526049419454902772554372, 14.04123015705678753926368803415, 14.87433917280849307581493720339, 15.21323141993701706596364405953, 16.39491758039977111732687643655, 17.33713828719823294144091840381, 18.17695076299861347051461328282, 18.94794102291873184222694225754, 19.76209777933774031179104537620, 20.554068038085733978917469313544, 20.95455608244677452337495235348

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