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Properties

Label 1-547-547.114-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: $\chi_{547} (114, \cdot )$
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

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

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