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Properties

Label 1-547-547.544-r0-0-0
Degree $1$
Conductor $547$
Sign $0.374 + 0.927i$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + 11-s + (0.623 + 0.781i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.900 + 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + 11-s + (0.623 + 0.781i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (0.623 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6996407997 + 0.4719465038i\)
\(L(\frac12)\) \(\approx\) \(0.6996407997 + 0.4719465038i\)
\(L(1)\) \(\approx\) \(0.6720519618 + 0.3087575096i\)
\(L(1)\) \(\approx\) \(0.6720519618 + 0.3087575096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + (-0.222 - 0.974i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.623 - 0.781i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−23.14929775961552359321280617749, −22.01451350809345913707870870699, −21.70757300229033760563592321109, −20.760265299439442528795664705798, −19.99746983019738688431928416369, −19.12441191464323206517334875258, −18.14408076221131585161506523260, −17.269118475922116062764965096238, −16.565832128945698247404044380562, −16.226448821317835384003394906496, −14.390964695769021481665089760641, −13.68070538560031528488078676114, −12.47342908205670340369111781042, −12.194303155989726221558023147790, −11.153693447721389690874487954208, −10.11939065537790676282952010072, −9.48555849360079394545909170039, −9.011107650102673301710610551034, −7.3373471673937392970095646010, −6.19017720121168260716631611248, −5.22334772367635119765844466841, −4.20738252161141248089696999245, −3.4950797120082089014134951119, −1.79651061822088717140663921981, −0.83171919596199538745079260802, 0.862442296680551206551507371376, 2.44412211642215281732984018043, 3.92607163327185728292418998730, 5.28567355128933284419877242957, 6.127485690330328067744017917539, 6.519453780174184634930435357743, 7.41101556229662993693807110586, 8.59293512813303594419815118774, 9.81611956266826088684822036302, 10.25930915940345436669883309387, 11.448278158711296250878744781326, 12.734314046678980073754984205943, 13.18301915872543535653377551924, 14.37528940039832368931652120036, 15.10837865212442085109699430121, 16.05870427910820676067644855569, 16.99734934533269389081174508920, 17.476211504776706260754451499729, 18.338293086956305174789657561333, 19.012203712808600699520256602936, 19.74188965527228582428303041813, 21.6436120251181837674550061354, 22.23006294706844731372923807702, 22.65466154529582593318986521095, 23.49449647634189629376147423067

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