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Properties

Label 1-1049-1049.1030-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.586 - 0.809i$
Analytic cond. $4.87153$
Root an. cond. $4.87153$
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  + (−0.935 + 0.352i)2-s + (−0.612 − 0.790i)3-s + (0.752 − 0.658i)4-s + (0.225 + 0.974i)5-s + (0.851 + 0.524i)6-s + (−0.995 + 0.0957i)7-s + (−0.472 + 0.881i)8-s + (−0.249 + 0.968i)9-s + (−0.554 − 0.832i)10-s + (−0.702 − 0.711i)11-s + (−0.981 − 0.190i)12-s + (−0.340 + 0.940i)13-s + (0.897 − 0.440i)14-s + (0.631 − 0.775i)15-s + (0.131 − 0.991i)16-s + (−0.272 + 0.962i)17-s + ⋯
L(s)  = 1  + (−0.935 + 0.352i)2-s + (−0.612 − 0.790i)3-s + (0.752 − 0.658i)4-s + (0.225 + 0.974i)5-s + (0.851 + 0.524i)6-s + (−0.995 + 0.0957i)7-s + (−0.472 + 0.881i)8-s + (−0.249 + 0.968i)9-s + (−0.554 − 0.832i)10-s + (−0.702 − 0.711i)11-s + (−0.981 − 0.190i)12-s + (−0.340 + 0.940i)13-s + (0.897 − 0.440i)14-s + (0.631 − 0.775i)15-s + (0.131 − 0.991i)16-s + (−0.272 + 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.586 - 0.809i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1030, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.586 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01488054946 + 0.02915460007i\)
\(L(\frac12)\) \(\approx\) \(0.01488054946 + 0.02915460007i\)
\(L(1)\) \(\approx\) \(0.4172075803 + 0.07032370793i\)
\(L(1)\) \(\approx\) \(0.4172075803 + 0.07032370793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.935 + 0.352i)T \)
3 \( 1 + (-0.612 - 0.790i)T \)
5 \( 1 + (0.225 + 0.974i)T \)
7 \( 1 + (-0.995 + 0.0957i)T \)
11 \( 1 + (-0.702 - 0.711i)T \)
13 \( 1 + (-0.340 + 0.940i)T \)
17 \( 1 + (-0.272 + 0.962i)T \)
19 \( 1 + (0.958 + 0.283i)T \)
23 \( 1 + (-0.887 + 0.461i)T \)
29 \( 1 + (0.971 + 0.237i)T \)
31 \( 1 + (-0.989 + 0.143i)T \)
37 \( 1 + (0.918 - 0.396i)T \)
41 \( 1 + (-0.958 - 0.283i)T \)
43 \( 1 + (-0.429 + 0.903i)T \)
47 \( 1 + (0.249 - 0.968i)T \)
53 \( 1 + (-0.985 - 0.167i)T \)
59 \( 1 + (-0.897 - 0.440i)T \)
61 \( 1 + (0.0838 + 0.996i)T \)
67 \( 1 + (-0.574 - 0.818i)T \)
71 \( 1 + (0.851 - 0.524i)T \)
73 \( 1 + (0.998 + 0.0479i)T \)
79 \( 1 + (0.0838 - 0.996i)T \)
83 \( 1 + (0.999 - 0.0239i)T \)
89 \( 1 + (-0.995 - 0.0957i)T \)
97 \( 1 + (0.782 + 0.622i)T \)
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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

−21.84642732792857158592034835759, −20.63397879780420930018329247755, −20.31266033414051575829997620661, −19.82849098998200144023746351064, −18.38644180257094598266058016360, −17.91131480343038001914978349335, −17.08812258076591980159566505824, −16.41384829332348924782486789869, −15.77121153039589239339176778836, −15.37432723963344236143300048829, −13.72573262347043962804027310244, −12.63600334122255018116638644622, −12.32641577919152966010343375271, −11.346971299871250033312100594842, −10.29733117968383162510029713270, −9.77139251688398935770980097229, −9.34131757537316524517684290578, −8.26701635841876271273797906721, −7.32111197922551897165103050181, −6.29485947084758700911544607297, −5.34647818944590020077841821325, −4.4952972044163101695013771047, −3.31988449932592402586846009264, −2.460394754981683555700725884743, −0.89876351867846060259957609047, 0.02540063498607444399789479204, 1.59057731616766154510159266460, 2.44605080120679387846689386400, 3.42520673013776456683170253671, 5.27232697168407806321989157353, 6.15757183497663822290720735396, 6.509674655685978213331077914098, 7.40367259360381125081801795933, 8.08517606238361324340539953194, 9.25581740021929949660666975855, 10.11965227663933460567761972028, 10.75513637299828048736468790651, 11.54802055256133618833859221111, 12.342022352502307572213008146024, 13.51340932620196059551661071697, 14.09702186338673776668216194977, 15.163703227424316126062647788525, 16.15681576863554675980807543529, 16.53470996931969222065072023881, 17.54028118701831375975806874499, 18.3019449474910294664635654237, 18.67015921153482754995612257925, 19.454018045195844555656543946484, 19.93272476418423137271937081898, 21.56414660494598138532129159689

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