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Properties

Label 1-1049-1049.1031-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.384 + 0.923i$
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  + (0.864 + 0.503i)2-s + (0.903 + 0.429i)3-s + (0.493 + 0.869i)4-s + (0.875 − 0.482i)5-s + (0.564 + 0.825i)6-s + (0.999 + 0.0359i)7-s + (−0.0119 + 0.999i)8-s + (0.631 + 0.775i)9-s + (0.999 + 0.0239i)10-s + (−0.995 − 0.0957i)11-s + (0.0718 + 0.997i)12-s + (0.752 − 0.658i)13-s + (0.845 + 0.534i)14-s + (0.998 − 0.0599i)15-s + (−0.513 + 0.857i)16-s + (−0.955 + 0.295i)17-s + ⋯
L(s)  = 1  + (0.864 + 0.503i)2-s + (0.903 + 0.429i)3-s + (0.493 + 0.869i)4-s + (0.875 − 0.482i)5-s + (0.564 + 0.825i)6-s + (0.999 + 0.0359i)7-s + (−0.0119 + 0.999i)8-s + (0.631 + 0.775i)9-s + (0.999 + 0.0239i)10-s + (−0.995 − 0.0957i)11-s + (0.0718 + 0.997i)12-s + (0.752 − 0.658i)13-s + (0.845 + 0.534i)14-s + (0.998 − 0.0599i)15-s + (−0.513 + 0.857i)16-s + (−0.955 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.384 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.670352306 + 2.448527688i\)
\(L(\frac12)\) \(\approx\) \(3.670352306 + 2.448527688i\)
\(L(1)\) \(\approx\) \(2.492805997 + 1.106820082i\)
\(L(1)\) \(\approx\) \(2.492805997 + 1.106820082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.864 + 0.503i)T \)
3 \( 1 + (0.903 + 0.429i)T \)
5 \( 1 + (0.875 - 0.482i)T \)
7 \( 1 + (0.999 + 0.0359i)T \)
11 \( 1 + (-0.995 - 0.0957i)T \)
13 \( 1 + (0.752 - 0.658i)T \)
17 \( 1 + (-0.955 + 0.295i)T \)
19 \( 1 + (-0.107 - 0.994i)T \)
23 \( 1 + (0.983 + 0.178i)T \)
29 \( 1 + (-0.767 - 0.640i)T \)
31 \( 1 + (-0.744 + 0.667i)T \)
37 \( 1 + (-0.237 - 0.971i)T \)
41 \( 1 + (-0.994 + 0.107i)T \)
43 \( 1 + (0.685 + 0.727i)T \)
47 \( 1 + (-0.775 + 0.631i)T \)
53 \( 1 + (-0.897 + 0.440i)T \)
59 \( 1 + (-0.534 - 0.845i)T \)
61 \( 1 + (0.225 - 0.974i)T \)
67 \( 1 + (-0.352 - 0.935i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (0.719 + 0.694i)T \)
79 \( 1 + (-0.225 - 0.974i)T \)
83 \( 1 + (0.374 + 0.927i)T \)
89 \( 1 + (-0.999 + 0.0359i)T \)
97 \( 1 + (-0.249 - 0.968i)T \)
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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

−21.07240558697174040633563556023, −20.824483207767995696493473063890, −20.22437571314927407220586725975, −18.88932540325484063441781499431, −18.545726238046057457162114299478, −17.89699435795679850922720022853, −16.611807929197558380510468905784, −15.401508490690025360326894351230, −14.833663452372494040294336854121, −14.19428241873170015742329342096, −13.38301602406094861776121922784, −13.133972609553067630191825736629, −11.9376677057213917965145985470, −10.99698264327061051547608371916, −10.41128461235725218549410519391, −9.37379549281785957292871649918, −8.56315741110935653426766334225, −7.39017439706185158621444422300, −6.69133411176388283145049880990, −5.69358237532849897474669824948, −4.80826798793845613484048776767, −3.76914308205691222564304909384, −2.824030165132894385455099374601, −1.95667653849035883341077926628, −1.488922022387590891595435196815, 1.675192574137604901842516050278, 2.45760593939395071184790448153, 3.35854627622201727443360908435, 4.58547099714558511011182479263, 5.026704889889043219294302057642, 5.87134465440643600025044855691, 7.08835672286793165004636723290, 8.02382114998421846585497177263, 8.59944963858439169649578044538, 9.37122960088595613026939169008, 10.84577472180648046690486841854, 11.02819079741456502246516971967, 12.81302694759848190985070292594, 13.07833713641644625081722673902, 13.82898592388189887368280843399, 14.535981386621914812394463559, 15.45467947912961429238421428629, 15.76228451212842366087612106339, 16.90838274618505320881767813878, 17.66165499470123331946746163673, 18.33050606565163370851144963128, 19.73507073500289660686040036431, 20.58871970119260912508124810918, 20.91725589833978285983482228702, 21.565834206531919310882984959746

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