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Properties

Label 1-1033-1033.101-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.937 + 0.347i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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  + (−0.872 + 0.489i)2-s + (−0.833 − 0.551i)3-s + (0.520 − 0.853i)4-s + (−0.983 + 0.181i)5-s + (0.997 + 0.0729i)6-s + (0.872 + 0.489i)7-s + (−0.0365 + 0.999i)8-s + (0.391 + 0.920i)9-s + (0.768 − 0.639i)10-s + (0.889 + 0.457i)11-s + (−0.905 + 0.424i)12-s + (0.920 − 0.391i)13-s − 14-s + (0.920 + 0.391i)15-s + (−0.457 − 0.889i)16-s + (0.976 + 0.217i)17-s + ⋯
L(s)  = 1  + (−0.872 + 0.489i)2-s + (−0.833 − 0.551i)3-s + (0.520 − 0.853i)4-s + (−0.983 + 0.181i)5-s + (0.997 + 0.0729i)6-s + (0.872 + 0.489i)7-s + (−0.0365 + 0.999i)8-s + (0.391 + 0.920i)9-s + (0.768 − 0.639i)10-s + (0.889 + 0.457i)11-s + (−0.905 + 0.424i)12-s + (0.920 − 0.391i)13-s − 14-s + (0.920 + 0.391i)15-s + (−0.457 − 0.889i)16-s + (0.976 + 0.217i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8144628087 + 0.1462461963i\)
\(L(\frac12)\) \(\approx\) \(0.8144628087 + 0.1462461963i\)
\(L(1)\) \(\approx\) \(0.6432373691 + 0.07480735244i\)
\(L(1)\) \(\approx\) \(0.6432373691 + 0.07480735244i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.872 + 0.489i)T \)
3 \( 1 + (-0.833 - 0.551i)T \)
5 \( 1 + (-0.983 + 0.181i)T \)
7 \( 1 + (0.872 + 0.489i)T \)
11 \( 1 + (0.889 + 0.457i)T \)
13 \( 1 + (0.920 - 0.391i)T \)
17 \( 1 + (0.976 + 0.217i)T \)
19 \( 1 + (0.744 - 0.667i)T \)
23 \( 1 + (0.0729 - 0.997i)T \)
29 \( 1 + (0.946 + 0.322i)T \)
31 \( 1 + (0.181 + 0.983i)T \)
37 \( 1 + (-0.457 - 0.889i)T \)
41 \( 1 + (-0.391 + 0.920i)T \)
43 \( 1 + (0.581 + 0.813i)T \)
47 \( 1 + (-0.999 - 0.0365i)T \)
53 \( 1 + (-0.983 + 0.181i)T \)
59 \( 1 + (-0.946 + 0.322i)T \)
61 \( 1 + (0.322 - 0.946i)T \)
67 \( 1 + (0.611 - 0.791i)T \)
71 \( 1 + (0.983 + 0.181i)T \)
73 \( 1 + (-0.889 - 0.457i)T \)
79 \( 1 + (0.889 + 0.457i)T \)
83 \( 1 + (0.0365 + 0.999i)T \)
89 \( 1 + (0.719 + 0.694i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−21.22331109395596080287905020362, −20.7611891860189926467484071950, −20.08655105043037795484391516033, −18.99930339092441371006438558555, −18.54796750610737517415939760913, −17.42836013130168128107522823191, −17.00079270782978485566855274831, −16.15774904913061243676528739517, −15.70685978766952542629286472198, −14.594317903488180737786408599890, −13.548492410025026893528712197154, −12.1415181229107978904943072035, −11.73454035018790296957167441483, −11.26676836031836726108801285692, −10.43744950657350050143262209461, −9.56943383581237803582489772067, −8.64763311882362439653433167846, −7.872303386541161057302666164897, −7.05793064768223164210451664951, −6.02448892676276834594180647630, −4.81467330676255010915761649587, −3.781164096978549963108064256801, −3.47377965857485907793575185369, −1.42772012349807771643301352159, −0.86685903147403889640549728202, 0.899461270057973733680951850591, 1.57313210514150635827452601905, 3.00289720397848549039551156311, 4.56088709439622612518476856249, 5.2693660603209783169662450955, 6.36346226510317611167634366716, 6.93583352721943483665787457801, 7.96801826045932922640593985195, 8.3022329930745338429728920024, 9.44642029884823165931286605382, 10.710368267228845556649666409641, 11.08643285156848160211538926669, 11.97178383662027873232278535896, 12.44855743895331995434741428793, 14.03863527959204260215813020335, 14.66041704674244265834789644007, 15.626459106347326952295055464238, 16.18093838283673659831220399046, 17.01273880098185416661411941377, 17.98078590841180253788004859562, 18.16903007216521592434145124657, 19.12139105253420664286816528531, 19.73778359774472732325791943313, 20.604716544262699287791052186619, 21.67916116890731995199554786206

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