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Properties

Label 1-1033-1033.1029-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.204 - 0.978i$
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  + (0.157 − 0.987i)2-s + (0.711 − 0.702i)3-s + (−0.950 − 0.311i)4-s + (0.276 + 0.961i)5-s + (−0.581 − 0.813i)6-s + (−0.934 − 0.357i)7-s + (−0.457 + 0.889i)8-s + (0.0121 − 0.999i)9-s + (0.992 − 0.121i)10-s + (0.915 + 0.402i)11-s + (−0.894 + 0.446i)12-s + (−0.0121 + 0.999i)13-s + (−0.5 + 0.866i)14-s + (0.872 + 0.489i)15-s + (0.806 + 0.591i)16-s + (−0.728 − 0.685i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.987i)2-s + (0.711 − 0.702i)3-s + (−0.950 − 0.311i)4-s + (0.276 + 0.961i)5-s + (−0.581 − 0.813i)6-s + (−0.934 − 0.357i)7-s + (−0.457 + 0.889i)8-s + (0.0121 − 0.999i)9-s + (0.992 − 0.121i)10-s + (0.915 + 0.402i)11-s + (−0.894 + 0.446i)12-s + (−0.0121 + 0.999i)13-s + (−0.5 + 0.866i)14-s + (0.872 + 0.489i)15-s + (0.806 + 0.591i)16-s + (−0.728 − 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146214986 - 1.410317756i\)
\(L(\frac12)\) \(\approx\) \(1.146214986 - 1.410317756i\)
\(L(1)\) \(\approx\) \(1.069838004 - 0.7568156257i\)
\(L(1)\) \(\approx\) \(1.069838004 - 0.7568156257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.157 - 0.987i)T \)
3 \( 1 + (0.711 - 0.702i)T \)
5 \( 1 + (0.276 + 0.961i)T \)
7 \( 1 + (-0.934 - 0.357i)T \)
11 \( 1 + (0.915 + 0.402i)T \)
13 \( 1 + (-0.0121 + 0.999i)T \)
17 \( 1 + (-0.728 - 0.685i)T \)
19 \( 1 + (0.561 + 0.827i)T \)
23 \( 1 + (0.413 - 0.910i)T \)
29 \( 1 + (0.0851 - 0.996i)T \)
31 \( 1 + (0.970 + 0.241i)T \)
37 \( 1 + (0.109 - 0.994i)T \)
41 \( 1 + (0.859 - 0.510i)T \)
43 \( 1 + (0.299 - 0.954i)T \)
47 \( 1 + (0.541 + 0.840i)T \)
53 \( 1 + (0.694 - 0.719i)T \)
59 \( 1 + (0.0851 + 0.996i)T \)
61 \( 1 + (0.905 - 0.424i)T \)
67 \( 1 + (0.985 + 0.169i)T \)
71 \( 1 + (0.276 - 0.961i)T \)
73 \( 1 + (-0.109 - 0.994i)T \)
79 \( 1 + (-0.806 + 0.591i)T \)
83 \( 1 + (-0.541 + 0.840i)T \)
89 \( 1 + (-0.520 + 0.853i)T \)
97 \( 1 + (-0.478 + 0.877i)T \)
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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

−21.88895707876810135602012229772, −21.299355288436801532010373651477, −19.962796188770257322230727252740, −19.76388257862729186970263621413, −18.69041780026671367749324724047, −17.451929886596212305032646820177, −17.02174580536788415632248861115, −16.00474685808141553585920659133, −15.69278042013955334157785689758, −14.91447190521017682386410388726, −13.93710059770066484958114785626, −13.183646230721147188354564074612, −12.80039922827722897894002149918, −11.52015425820069824650745931518, −10.08820809080929503592937388818, −9.47199852110113828724883951815, −8.79604050339713168246128836552, −8.30875818823431837917282296160, −7.1489156961711039971896099886, −6.09726060740623599517364158785, −5.36211516807362624338950164089, −4.49611757082117446686652675370, −3.61544220931191903941326303670, −2.77917105111889781335838459651, −1.02704321893103347782389753785, 0.85443811449816984513255801348, 2.11852045894981353277154441218, 2.63269284220770128644061013097, 3.68319833091794221071479193527, 4.23816030974720240438512909967, 5.961717988299960374507861370090, 6.69644649877214808499133489642, 7.36981929330476593186309069708, 8.72373436651584084094555428362, 9.47178465033313309438851123554, 9.95886915247226718263148174134, 11.059799039676136981514800056646, 11.92705727930783513691878382609, 12.55030143069558582651362898113, 13.601528242890270964646886380703, 14.011068150176216740566832400538, 14.58097521361787835344008150315, 15.63036878053573684595034823372, 16.95400591914030300038870591564, 17.80465272334426110586391994880, 18.52150125809594919752978800129, 19.28950091493778785806342446831, 19.499295930648608337707322227520, 20.55854003690910923449302410762, 21.124810414833087823546874912274

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