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Properties

Label 1-547-547.4-r0-0-0
Degree $1$
Conductor $547$
Sign $0.999 + 0.0363i$
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  + (0.999 + 0.0230i)2-s + (−0.900 − 0.433i)3-s + (0.998 + 0.0460i)4-s + (−0.558 − 0.829i)5-s + (−0.890 − 0.454i)6-s + (−0.717 + 0.696i)7-s + (0.997 + 0.0689i)8-s + (0.623 + 0.781i)9-s + (−0.539 − 0.842i)10-s + (−0.0402 + 0.999i)11-s + (−0.880 − 0.474i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (0.143 + 0.989i)15-s + (0.995 + 0.0919i)16-s + (−0.944 − 0.327i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0230i)2-s + (−0.900 − 0.433i)3-s + (0.998 + 0.0460i)4-s + (−0.558 − 0.829i)5-s + (−0.890 − 0.454i)6-s + (−0.717 + 0.696i)7-s + (0.997 + 0.0689i)8-s + (0.623 + 0.781i)9-s + (−0.539 − 0.842i)10-s + (−0.0402 + 0.999i)11-s + (−0.880 − 0.474i)12-s + (0.955 + 0.294i)13-s + (−0.733 + 0.680i)14-s + (0.143 + 0.989i)15-s + (0.995 + 0.0919i)16-s + (−0.944 − 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.999 + 0.0363i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.999 + 0.0363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.743903977 + 0.03170827643i\)
\(L(\frac12)\) \(\approx\) \(1.743903977 + 0.03170827643i\)
\(L(1)\) \(\approx\) \(1.377262028 - 0.07782412872i\)
\(L(1)\) \(\approx\) \(1.377262028 - 0.07782412872i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.999 + 0.0230i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.558 - 0.829i)T \)
7 \( 1 + (-0.717 + 0.696i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.944 - 0.327i)T \)
19 \( 1 + (0.548 - 0.835i)T \)
23 \( 1 + (0.895 + 0.444i)T \)
29 \( 1 + (0.509 + 0.860i)T \)
31 \( 1 + (0.770 + 0.636i)T \)
37 \( 1 + (0.211 - 0.977i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.980 - 0.194i)T \)
47 \( 1 + (0.987 - 0.160i)T \)
53 \( 1 + (0.785 - 0.618i)T \)
59 \( 1 + (0.692 - 0.721i)T \)
61 \( 1 + (0.641 - 0.767i)T \)
67 \( 1 + (0.0287 + 0.999i)T \)
71 \( 1 + (0.924 - 0.381i)T \)
73 \( 1 + (0.143 - 0.989i)T \)
79 \( 1 + (0.322 + 0.946i)T \)
83 \( 1 + (0.428 + 0.903i)T \)
89 \( 1 + (-0.985 + 0.171i)T \)
97 \( 1 + (-0.965 + 0.261i)T \)
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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

−23.16572263476715422810979548638, −22.59694734020498692567313054889, −22.059237443152238039494523039695, −21.06230782024014643036982115291, −20.28969637619059198538487283046, −19.20412408539942653827046868820, −18.50069297034514048059353257420, −17.14810734819687125565098510026, −16.39710138028060821295651821184, −15.661530516469063333013158572346, −15.11358536309769249878873179916, −13.81508533539181588837688498834, −13.25916245990543218463978562318, −12.08393029396339251777739198507, −11.310082848877427312773246227522, −10.68121661577819266463027195205, −10.05230802827502927817489233031, −8.31274442329351424928500443537, −7.04844000827164457028582081995, −6.381846139436825750312578799343, −5.737571668217741068439684095255, −4.3269920379688609432633959426, −3.70604241775654551497089887516, −2.88354687780391116140820148310, −0.9059077348655025040872335740, 1.164476926517023306915237861649, 2.390871463373962187744497392602, 3.7475340139228877565241617809, 4.87188527314225198665896500835, 5.31659278186757708962967470383, 6.63606346905716770216521831858, 7.02550768769805340955960681297, 8.38862666639316514618276927295, 9.53672563286573774587171339615, 10.89067954319882980418522162191, 11.64977064981604749214829844151, 12.301663808437179339520822262645, 13.03658128011159138787867530215, 13.56075505368608543131074045663, 15.18386990602832448113291392330, 15.815003602667665882322385466830, 16.29155362991740805719402303365, 17.36492677247483361876740629531, 18.30453823391488140347950028590, 19.45905238554040900009448504959, 20.02711667759756984812032708849, 21.09748108795651331082934199283, 21.92200349699906077874138639298, 22.72389507387988112873495557811, 23.36215950614064775724659546830

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