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Properties

Label 1-547-547.171-r0-0-0
Degree $1$
Conductor $547$
Sign $0.173 - 0.984i$
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  + (0.998 − 0.0460i)2-s + (0.623 − 0.781i)3-s + (0.995 − 0.0919i)4-s + (−0.376 − 0.926i)5-s + (0.586 − 0.809i)6-s + (0.0287 + 0.999i)7-s + (0.990 − 0.137i)8-s + (−0.222 − 0.974i)9-s + (−0.418 − 0.908i)10-s + (−0.996 + 0.0804i)11-s + (0.548 − 0.835i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (−0.958 − 0.283i)15-s + (0.983 − 0.183i)16-s + (0.785 − 0.618i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0460i)2-s + (0.623 − 0.781i)3-s + (0.995 − 0.0919i)4-s + (−0.376 − 0.926i)5-s + (0.586 − 0.809i)6-s + (0.0287 + 0.999i)7-s + (0.990 − 0.137i)8-s + (−0.222 − 0.974i)9-s + (−0.418 − 0.908i)10-s + (−0.996 + 0.0804i)11-s + (0.548 − 0.835i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (−0.958 − 0.283i)15-s + (0.983 − 0.183i)16-s + (0.785 − 0.618i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.309050988 - 1.938713125i\)
\(L(\frac12)\) \(\approx\) \(2.309050988 - 1.938713125i\)
\(L(1)\) \(\approx\) \(1.987121635 - 0.8893160048i\)
\(L(1)\) \(\approx\) \(1.987121635 - 0.8893160048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.998 - 0.0460i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.376 - 0.926i)T \)
7 \( 1 + (0.0287 + 0.999i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.785 - 0.618i)T \)
19 \( 1 + (-0.397 + 0.917i)T \)
23 \( 1 + (0.605 - 0.795i)T \)
29 \( 1 + (-0.479 - 0.877i)T \)
31 \( 1 + (0.188 - 0.982i)T \)
37 \( 1 + (-0.910 + 0.413i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.924 - 0.381i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (0.233 + 0.972i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (-0.177 + 0.984i)T \)
67 \( 1 + (-0.998 - 0.0575i)T \)
71 \( 1 + (0.709 + 0.705i)T \)
73 \( 1 + (-0.958 + 0.283i)T \)
79 \( 1 + (-0.792 - 0.609i)T \)
83 \( 1 + (-0.632 - 0.774i)T \)
89 \( 1 + (0.940 + 0.338i)T \)
97 \( 1 + (0.863 + 0.504i)T \)
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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

−23.46286695356962749589554803483, −22.80702894156129960764209651019, −21.76361786528947352705392214725, −21.195625809575647717961915338695, −20.49197017636061614790739413760, −19.52668320153678011367594901072, −18.96361966512700327584621193635, −17.47217826904599059999429518834, −16.39062720471312237263479618906, −15.70940351718220401370045577114, −15.051726764969024549248429487011, −14.10463828747762204354162099966, −13.69983126500693029448150814111, −12.68219539913839300340697144831, −11.158798619387073094876088382151, −10.82675992975547080732394834092, −10.0976779895749648087963933626, −8.57323650341400689907073375331, −7.52625901725943436697425227097, −6.91858961663162798662582390317, −5.58543863820512894377806166720, −4.5704674239730498850155436119, −3.580032574253390872734887840670, −3.183821451961082570314937280445, −1.87130629060698603437726738720, 1.13471040273619889380740294795, 2.326381766286294117216806448853, 3.142251235529028049252020260251, 4.29173697481699171173179463906, 5.521456534741074817989539876392, 6.01806537524714280789517100276, 7.50688453603223083574492495969, 8.09670739683805960692945419035, 8.984392323985518909488366651274, 10.32279164009935985580651424718, 11.681926158929198365257979966895, 12.24380547986002134492155368162, 12.99469890941751460118574000157, 13.51462083914875348796389619798, 14.710669054236133997656326513165, 15.39135508328092241811262752137, 16.091121622256320814274590584687, 17.17971016369863555516930808658, 18.66661357730191396410010829681, 18.863471239752697619290659852583, 20.181566964207168016534624195381, 20.88207374761069576016642845461, 21.06451039878507384551956909014, 22.68711025390059999347213102831, 23.21455198156181349328980438730

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