Properties

Label 1-547-547.488-r0-0-0
Degree $1$
Conductor $547$
Sign $0.983 - 0.180i$
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  + (−0.919 + 0.391i)2-s + 3-s + (0.692 − 0.721i)4-s + (0.987 + 0.160i)5-s + (−0.919 + 0.391i)6-s + (0.799 − 0.600i)7-s + (−0.354 + 0.935i)8-s + 9-s + (−0.970 + 0.239i)10-s + (−0.996 − 0.0804i)11-s + (0.692 − 0.721i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.987 + 0.160i)15-s + (−0.0402 − 0.999i)16-s + (0.428 − 0.903i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.391i)2-s + 3-s + (0.692 − 0.721i)4-s + (0.987 + 0.160i)5-s + (−0.919 + 0.391i)6-s + (0.799 − 0.600i)7-s + (−0.354 + 0.935i)8-s + 9-s + (−0.970 + 0.239i)10-s + (−0.996 − 0.0804i)11-s + (0.692 − 0.721i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (0.987 + 0.160i)15-s + (−0.0402 − 0.999i)16-s + (0.428 − 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.983 - 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.626057821 - 0.1479221063i\)
\(L(\frac12)\) \(\approx\) \(1.626057821 - 0.1479221063i\)
\(L(1)\) \(\approx\) \(1.223836237 + 0.01985722233i\)
\(L(1)\) \(\approx\) \(1.223836237 + 0.01985722233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.919 + 0.391i)T \)
3 \( 1 + T \)
5 \( 1 + (0.987 + 0.160i)T \)
7 \( 1 + (0.799 - 0.600i)T \)
11 \( 1 + (-0.996 - 0.0804i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.428 - 0.903i)T \)
19 \( 1 + (-0.0402 + 0.999i)T \)
23 \( 1 + (-0.200 - 0.979i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.885 + 0.464i)T \)
37 \( 1 + (-0.200 + 0.979i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (0.948 - 0.316i)T \)
53 \( 1 + (-0.632 + 0.774i)T \)
59 \( 1 + (-0.0402 - 0.999i)T \)
61 \( 1 + (-0.919 + 0.391i)T \)
67 \( 1 + (0.278 + 0.960i)T \)
71 \( 1 + (-0.845 - 0.534i)T \)
73 \( 1 + (0.987 - 0.160i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.632 + 0.774i)T \)
89 \( 1 + (0.120 + 0.992i)T \)
97 \( 1 + (-0.996 + 0.0804i)T \)
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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

−23.9826663854892762079723281965, −21.85269123383785066612471639345, −21.47172119593226067412774695975, −20.92871628863674101234919603459, −20.07073894717141466469780614021, −19.11770618709360601653660387878, −18.4597446548167336448861885822, −17.69809184791544947324785439615, −16.92938557483372300764726971063, −15.73832429728742336444713844903, −15.02196670648687345227973366671, −14.00233112107491148623713525346, −13.08973084501498600937741583360, −12.299923594355818966220421006919, −11.1029396466186093976239612201, −10.1768344894994617510721027399, −9.37140840481091560067824444404, −8.753635404057203773352604520794, −7.87482536263550195089147253931, −7.033766570801980501389116280626, −5.67459988621665648827464922147, −4.45121903154863381246032263106, −2.99421845054456480354354007423, −2.138493841718534342172991999253, −1.56296381113818664840116627142, 1.113186095868227106161626796898, 2.20742995561209738838157605354, 2.96874143553880612701275138738, 4.75340093604230255083445823009, 5.66726021345034075691909033467, 6.93626092408782938364701275764, 7.81788178359533590552338246799, 8.296591288303112147877136806690, 9.49815917222877121602776532508, 10.22646637010587438927675444104, 10.66905623457320627773860413131, 12.22872872010395123249494927965, 13.46840530634034487679328264567, 14.16259542793968751058686210578, 14.825847874464802156124580411213, 15.71726428222030877335296308094, 16.75925405284706006335472549788, 17.5609693034515967892744578831, 18.422349149764433460029343000427, 18.83341246099198895693976196755, 20.21643265953150299646289117331, 20.664515612040476354851745288290, 21.155542408059879436590741243617, 22.5792040558135871586082176899, 23.70532887310790662567297715661

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