Properties

Label 1-547-547.24-r0-0-0
Degree $1$
Conductor $547$
Sign $0.686 - 0.727i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.256 − 0.966i)2-s + (−0.900 + 0.433i)3-s + (−0.868 − 0.495i)4-s + (−0.700 − 0.713i)5-s + (0.188 + 0.982i)6-s + (−0.999 + 0.0345i)7-s + (−0.700 + 0.713i)8-s + (0.623 − 0.781i)9-s + (−0.868 + 0.495i)10-s + (−0.748 + 0.663i)11-s + (0.997 + 0.0689i)12-s + (−0.222 + 0.974i)13-s + (−0.222 + 0.974i)14-s + (0.940 + 0.338i)15-s + (0.509 + 0.860i)16-s + (−0.985 + 0.171i)17-s + ⋯
L(s)  = 1  + (0.256 − 0.966i)2-s + (−0.900 + 0.433i)3-s + (−0.868 − 0.495i)4-s + (−0.700 − 0.713i)5-s + (0.188 + 0.982i)6-s + (−0.999 + 0.0345i)7-s + (−0.700 + 0.713i)8-s + (0.623 − 0.781i)9-s + (−0.868 + 0.495i)10-s + (−0.748 + 0.663i)11-s + (0.997 + 0.0689i)12-s + (−0.222 + 0.974i)13-s + (−0.222 + 0.974i)14-s + (0.940 + 0.338i)15-s + (0.509 + 0.860i)16-s + (−0.985 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.686 - 0.727i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.686 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4559491414 - 0.1965488478i\)
\(L(\frac12)\) \(\approx\) \(0.4559491414 - 0.1965488478i\)
\(L(1)\) \(\approx\) \(0.5220683089 - 0.2361379923i\)
\(L(1)\) \(\approx\) \(0.5220683089 - 0.2361379923i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.256 - 0.966i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.700 - 0.713i)T \)
7 \( 1 + (-0.999 + 0.0345i)T \)
11 \( 1 + (-0.748 + 0.663i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.985 + 0.171i)T \)
19 \( 1 + (0.990 - 0.137i)T \)
23 \( 1 + (0.449 - 0.893i)T \)
29 \( 1 + (-0.792 - 0.609i)T \)
31 \( 1 + (-0.0862 - 0.996i)T \)
37 \( 1 + (-0.418 + 0.908i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.154 - 0.987i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (0.940 + 0.338i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (0.915 - 0.402i)T \)
67 \( 1 + (0.997 + 0.0689i)T \)
71 \( 1 + (-0.952 - 0.305i)T \)
73 \( 1 + (0.940 - 0.338i)T \)
79 \( 1 + (-0.154 - 0.987i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (0.915 + 0.402i)T \)
97 \( 1 + (0.813 - 0.582i)T \)
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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

−23.29982718260365957081244924151, −22.85829934994525676634292643821, −22.25574795895177525829002556767, −21.509218489136968284643623521308, −19.83528462731371913602661643103, −19.06401693499081157217573839192, −18.1446327060240533076646507530, −17.74850112471250175343481368157, −16.444208540536894734281819741645, −15.927719795794426016526385296231, −15.366573922241488517799351050640, −14.106895875926853050441363258514, −13.12151534582328903321450064404, −12.665166542616183763827285448571, −11.48980046064402753195946354307, −10.635574684099925023568369486286, −9.58795019451109865384536321866, −8.22995169672747307857127869990, −7.335023541716521462003625096118, −6.83151767879002886469407785543, −5.78051472773623644284299519409, −5.12345336904653425242275924920, −3.70226905459362092850408628292, −2.88169861980971111449026948293, −0.49766062157787769292509467590, 0.6326796178963144962353972560, 2.20212904753563592441554139229, 3.58250842601264105944145137226, 4.418425773146806772180943901879, 5.065102987325823473974034325074, 6.19593527983191946959200302255, 7.35024630852388628003999851196, 8.9082917769245888691314171738, 9.53715119978518149859000360605, 10.380455022453336708615274108064, 11.405623448864575883597502654321, 11.985963219238194865345271931338, 12.83546085551129443449089670757, 13.37922449251020620856010427036, 14.97091094202466805777183467883, 15.6780069646252006788677649712, 16.52851486105760215304451752154, 17.36412164493751566088025866079, 18.46881588535770602492179975289, 19.10685499703508742225757817585, 20.1867417956481980562098554800, 20.67661810573280332172743823277, 21.64820173145839412352933618723, 22.60119926249151228905508497795, 22.86201338078922583659605683105

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