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Properties

Label 1-1049-1049.1020-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.00871 - 0.999i$
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  + (−0.295 − 0.955i)2-s + (0.958 − 0.283i)3-s + (−0.825 + 0.564i)4-s + (−0.997 − 0.0718i)5-s + (−0.554 − 0.832i)6-s + (−0.429 + 0.903i)7-s + (0.782 + 0.622i)8-s + (0.838 − 0.544i)9-s + (0.225 + 0.974i)10-s + (0.612 − 0.790i)11-s + (−0.631 + 0.775i)12-s + (−0.851 − 0.524i)13-s + (0.989 + 0.143i)14-s + (−0.976 + 0.214i)15-s + (0.363 − 0.931i)16-s + (−0.472 + 0.881i)17-s + ⋯
L(s)  = 1  + (−0.295 − 0.955i)2-s + (0.958 − 0.283i)3-s + (−0.825 + 0.564i)4-s + (−0.997 − 0.0718i)5-s + (−0.554 − 0.832i)6-s + (−0.429 + 0.903i)7-s + (0.782 + 0.622i)8-s + (0.838 − 0.544i)9-s + (0.225 + 0.974i)10-s + (0.612 − 0.790i)11-s + (−0.631 + 0.775i)12-s + (−0.851 − 0.524i)13-s + (0.989 + 0.143i)14-s + (−0.976 + 0.214i)15-s + (0.363 − 0.931i)16-s + (−0.472 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00871 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00871 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.00871 - 0.999i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1020, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.00871 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8835640785 - 0.8912939051i\)
\(L(\frac12)\) \(\approx\) \(0.8835640785 - 0.8912939051i\)
\(L(1)\) \(\approx\) \(0.8592087820 - 0.4726233589i\)
\(L(1)\) \(\approx\) \(0.8592087820 - 0.4726233589i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.295 - 0.955i)T \)
3 \( 1 + (0.958 - 0.283i)T \)
5 \( 1 + (-0.997 - 0.0718i)T \)
7 \( 1 + (-0.429 + 0.903i)T \)
11 \( 1 + (0.612 - 0.790i)T \)
13 \( 1 + (-0.851 - 0.524i)T \)
17 \( 1 + (-0.472 + 0.881i)T \)
19 \( 1 + (0.971 + 0.237i)T \)
23 \( 1 + (-0.797 - 0.603i)T \)
29 \( 1 + (0.318 + 0.948i)T \)
31 \( 1 + (-0.992 + 0.119i)T \)
37 \( 1 + (0.649 - 0.760i)T \)
41 \( 1 + (0.971 + 0.237i)T \)
43 \( 1 + (-0.107 + 0.994i)T \)
47 \( 1 + (0.838 - 0.544i)T \)
53 \( 1 + (0.927 - 0.374i)T \)
59 \( 1 + (0.989 - 0.143i)T \)
61 \( 1 + (0.981 + 0.190i)T \)
67 \( 1 + (0.272 - 0.962i)T \)
71 \( 1 + (-0.554 + 0.832i)T \)
73 \( 1 + (0.534 - 0.845i)T \)
79 \( 1 + (0.981 - 0.190i)T \)
83 \( 1 + (-0.875 - 0.482i)T \)
89 \( 1 + (-0.429 - 0.903i)T \)
97 \( 1 + (0.0359 - 0.999i)T \)
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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

−22.21088374075828700972046360968, −20.57434637511679901232759959777, −19.90376244046911744319619623885, −19.55323967233688803155063417512, −18.71368038153225959313556801225, −17.761767030057167648814670938065, −16.816847357669967211475704880320, −16.08456325292818459692356890535, −15.54461487422037448799343239139, −14.73700611074908288934194892049, −14.06711530871066049569258163211, −13.442261246975053495622834069871, −12.35784973722532230178879067622, −11.304973287500903995450486272709, −10.03966852368413736326412404451, −9.59723858927246843314022077377, −8.83150661036221641785433932599, −7.62274108187074976749493721936, −7.38678194312244766087180344131, −6.71047020389125290058193565858, −5.10244703028265412984171677260, −4.17929401824390667441270300585, −3.85502731342565219257162509116, −2.430283608392264572838917321241, −0.921437213826536389611689851412, 0.70674905453610373476386112545, 1.98417787225792933953734315514, 2.90913345227717551205316854408, 3.547971747246356718463063151568, 4.33151469096632688034747283050, 5.62794672297161113303961494850, 6.991239847525904458106736708598, 7.915892064193052021084069864104, 8.555260622408482234612251773356, 9.15189471777609085745000844500, 10.01071914274677487815692021731, 11.05559226430020836240119788639, 11.97029115646845132924596202346, 12.52452611285178950605844610264, 13.10385921776865217879709062030, 14.3566110027744073858164358599, 14.76573205428804495060746560016, 15.917358584566375218936409736229, 16.539951839729181962862716775434, 17.93539701065536656876609290667, 18.43887154828666658956831927588, 19.31765861773670787201034641865, 19.75503866283587126603325767032, 20.13145741383883765088737426525, 21.28231039180617415327333481590

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