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Properties

Label 1-547-547.140-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.997 - 0.0720i$
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  + (−0.991 − 0.126i)2-s + (0.623 + 0.781i)3-s + (0.968 + 0.250i)4-s + (0.785 + 0.618i)5-s + (−0.519 − 0.854i)6-s + (−0.890 − 0.454i)7-s + (−0.928 − 0.370i)8-s + (−0.222 + 0.974i)9-s + (−0.700 − 0.713i)10-s + (−0.845 + 0.534i)11-s + (0.407 + 0.913i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.00575 + 0.999i)15-s + (0.874 + 0.484i)16-s + (−0.965 − 0.261i)17-s + ⋯
L(s)  = 1  + (−0.991 − 0.126i)2-s + (0.623 + 0.781i)3-s + (0.968 + 0.250i)4-s + (0.785 + 0.618i)5-s + (−0.519 − 0.854i)6-s + (−0.890 − 0.454i)7-s + (−0.928 − 0.370i)8-s + (−0.222 + 0.974i)9-s + (−0.700 − 0.713i)10-s + (−0.845 + 0.534i)11-s + (0.407 + 0.913i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.00575 + 0.999i)15-s + (0.874 + 0.484i)16-s + (−0.965 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.997 - 0.0720i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.997 - 0.0720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01325140048 + 0.3675008246i\)
\(L(\frac12)\) \(\approx\) \(0.01325140048 + 0.3675008246i\)
\(L(1)\) \(\approx\) \(0.5897560584 + 0.2443961498i\)
\(L(1)\) \(\approx\) \(0.5897560584 + 0.2443961498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.991 - 0.126i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.785 + 0.618i)T \)
7 \( 1 + (-0.890 - 0.454i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (-0.965 - 0.261i)T \)
19 \( 1 + (-0.667 - 0.744i)T \)
23 \( 1 + (-0.819 + 0.572i)T \)
29 \( 1 + (-0.832 + 0.553i)T \)
31 \( 1 + (-0.792 - 0.609i)T \)
37 \( 1 + (-0.376 + 0.926i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.880 + 0.474i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (0.863 - 0.504i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (0.0976 + 0.995i)T \)
67 \( 1 + (0.586 - 0.809i)T \)
71 \( 1 + (0.548 + 0.835i)T \)
73 \( 1 + (0.00575 - 0.999i)T \)
79 \( 1 + (0.851 + 0.524i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (0.813 + 0.582i)T \)
97 \( 1 + (0.993 + 0.114i)T \)
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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

−23.337420489865641447169285581646, −21.78213237577274543801189760754, −21.0696783293637504672711100894, −20.21831435188496829038439652250, −19.46599564305464181970843791605, −18.60359603435947377686220000436, −18.22779775601568774745840884719, −17.08986096277905776782055146878, −16.37140317667331162781761201720, −15.53288798213803428829794078689, −14.4277922330350360950104628112, −13.451544380654768356893969796574, −12.69692697609688364477349709153, −11.88729863068780561826491747476, −10.554764898458322023151813568499, −9.6519844955830818431572201041, −8.76770955523823075999304001875, −8.45235349351251571910539330821, −7.10178845970964022692514645103, −6.30406770816600978271494163866, −5.61318353306745371169447347344, −3.674637330844826180840252356584, −2.238402336390289345140286948472, −1.93663312027097231682912914829, −0.21091539102175037137697932596, 1.9873317969448136390992934829, 2.78426472752259315669551322059, 3.595988049772578487586770786894, 5.16935432122296357694035202768, 6.36165823778682381950652007263, 7.27775733677678073561406614466, 8.20694741237715023279424927897, 9.337576996432442701181034478670, 9.88327803288003438690776794454, 10.53521362809913571460516153937, 11.21157948918105514236537892565, 12.99062616906648176223152815634, 13.38150408972676597902475208326, 14.85901175082860928805080920364, 15.40050093309111331047820034379, 16.212702701402408213654656202326, 17.15692857137721076112991184369, 17.95735956170352652065562757611, 18.73333170607051716842504479665, 19.89359387430152184843836845987, 20.16537351165640263720129081851, 21.14582800279244548763029295881, 21.99510185189091205102231123164, 22.64036578370690274279053533521, 24.00090257236240551957778982639

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