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Properties

Label 1-1049-1049.110-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.100 + 0.994i$
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  + (−0.797 + 0.603i)2-s + (0.927 + 0.374i)3-s + (0.272 − 0.962i)4-s + (0.995 − 0.0957i)5-s + (−0.965 + 0.260i)6-s + (−0.897 − 0.440i)7-s + (0.363 + 0.931i)8-s + (0.719 + 0.694i)9-s + (−0.736 + 0.676i)10-s + (−0.985 − 0.167i)11-s + (0.612 − 0.790i)12-s + (−0.951 − 0.306i)13-s + (0.981 − 0.190i)14-s + (0.958 + 0.283i)15-s + (−0.851 − 0.524i)16-s + (0.131 + 0.991i)17-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)2-s + (0.927 + 0.374i)3-s + (0.272 − 0.962i)4-s + (0.995 − 0.0957i)5-s + (−0.965 + 0.260i)6-s + (−0.897 − 0.440i)7-s + (0.363 + 0.931i)8-s + (0.719 + 0.694i)9-s + (−0.736 + 0.676i)10-s + (−0.985 − 0.167i)11-s + (0.612 − 0.790i)12-s + (−0.951 − 0.306i)13-s + (0.981 − 0.190i)14-s + (0.958 + 0.283i)15-s + (−0.851 − 0.524i)16-s + (0.131 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8893213023 + 0.9838455812i\)
\(L(\frac12)\) \(\approx\) \(0.8893213023 + 0.9838455812i\)
\(L(1)\) \(\approx\) \(0.9234753918 + 0.4144490500i\)
\(L(1)\) \(\approx\) \(0.9234753918 + 0.4144490500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.797 + 0.603i)T \)
3 \( 1 + (0.927 + 0.374i)T \)
5 \( 1 + (0.995 - 0.0957i)T \)
7 \( 1 + (-0.897 - 0.440i)T \)
11 \( 1 + (-0.985 - 0.167i)T \)
13 \( 1 + (-0.951 - 0.306i)T \)
17 \( 1 + (0.131 + 0.991i)T \)
19 \( 1 + (-0.202 + 0.979i)T \)
23 \( 1 + (0.649 - 0.760i)T \)
29 \( 1 + (0.908 + 0.418i)T \)
31 \( 1 + (-0.631 + 0.775i)T \)
37 \( 1 + (0.407 + 0.913i)T \)
41 \( 1 + (-0.202 + 0.979i)T \)
43 \( 1 + (0.989 - 0.143i)T \)
47 \( 1 + (0.719 + 0.694i)T \)
53 \( 1 + (-0.0119 - 0.999i)T \)
59 \( 1 + (0.981 + 0.190i)T \)
61 \( 1 + (-0.702 - 0.711i)T \)
67 \( 1 + (-0.155 + 0.987i)T \)
71 \( 1 + (-0.965 - 0.260i)T \)
73 \( 1 + (0.225 + 0.974i)T \)
79 \( 1 + (-0.702 + 0.711i)T \)
83 \( 1 + (0.782 - 0.622i)T \)
89 \( 1 + (-0.897 + 0.440i)T \)
97 \( 1 + (0.998 - 0.0479i)T \)
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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

−21.18675260888384309225101602477, −20.48523734456736095031113013380, −19.606729873343802440357452072062, −19.073772430166027743094350073453, −18.30874259041319088296749302876, −17.752147169864742792355007828177, −16.84629860447707066959154197897, −15.83633894305379195567966581439, −15.18539272601977764149192523912, −13.94051369261101264039378594294, −13.274989642550877838397210108905, −12.715398975226389640672346806783, −11.90639846403008838897467562875, −10.648413036061922439313324196637, −9.83636515079874666848973644200, −9.290446991403608833736530525, −8.81639549358817320180803027732, −7.36869980544821968629586986555, −7.180713659842589433789669504872, −5.92600399281339896775984835292, −4.6559761426683445707297789583, −3.25154140580097898364100630676, −2.48093014594225361773516973046, −2.21627458457688077300081911652, −0.65932774826316203383326254668, 1.2258086250259669641341053235, 2.36682060039685440374455853310, 3.07100826972200063306305783531, 4.54841612365747622956656639277, 5.44883760863378272238411173254, 6.3821084488568036774275783032, 7.24675504961340480268963590022, 8.14320231765120157041893237273, 8.83559855892236711195867221635, 9.77582424195491490214635449912, 10.25839744704698518833718061231, 10.65591075246610122122584042225, 12.639429873652997707548403471472, 13.12426380348571626485658728506, 14.19992805534097675148099090882, 14.624449538445959864106599161919, 15.5645195832989557253500034035, 16.39123056461493000330439237631, 16.88176706073781043120358257544, 17.81639108021216346240932766040, 18.73432296973054407229317414163, 19.28481005555441914199904621092, 20.116875837856641751398564786980, 20.7576607692030694167845604719, 21.59178326029513246327924604300

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