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Properties

Label 1-547-547.49-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.288 - 0.957i$
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  + (−0.717 + 0.696i)2-s + (0.623 − 0.781i)3-s + (0.0287 − 0.999i)4-s + (−0.976 − 0.216i)5-s + (0.0976 + 0.995i)6-s + (0.641 − 0.767i)7-s + (0.675 + 0.736i)8-s + (−0.222 − 0.974i)9-s + (0.851 − 0.524i)10-s + (−0.845 + 0.534i)11-s + (−0.763 − 0.645i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (−0.778 + 0.627i)15-s + (−0.998 − 0.0575i)16-s + (0.983 − 0.183i)17-s + ⋯
L(s)  = 1  + (−0.717 + 0.696i)2-s + (0.623 − 0.781i)3-s + (0.0287 − 0.999i)4-s + (−0.976 − 0.216i)5-s + (0.0976 + 0.995i)6-s + (0.641 − 0.767i)7-s + (0.675 + 0.736i)8-s + (−0.222 − 0.974i)9-s + (0.851 − 0.524i)10-s + (−0.845 + 0.534i)11-s + (−0.763 − 0.645i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (−0.778 + 0.627i)15-s + (−0.998 − 0.0575i)16-s + (0.983 − 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.288 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5152536647 - 0.6931856898i\)
\(L(\frac12)\) \(\approx\) \(0.5152536647 - 0.6931856898i\)
\(L(1)\) \(\approx\) \(0.7588303629 - 0.2329180452i\)
\(L(1)\) \(\approx\) \(0.7588303629 - 0.2329180452i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.717 + 0.696i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.976 - 0.216i)T \)
7 \( 1 + (0.641 - 0.767i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.983 - 0.183i)T \)
19 \( 1 + (0.166 - 0.986i)T \)
23 \( 1 + (-0.958 - 0.283i)T \)
29 \( 1 + (0.990 - 0.137i)T \)
31 \( 1 + (-0.418 + 0.908i)T \)
37 \( 1 + (-0.0632 - 0.997i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.267 - 0.963i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (0.932 + 0.359i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (-0.519 - 0.854i)T \)
67 \( 1 + (-0.177 + 0.984i)T \)
71 \( 1 + (-0.857 - 0.514i)T \)
73 \( 1 + (-0.778 - 0.627i)T \)
79 \( 1 + (-0.700 + 0.713i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (-0.479 - 0.877i)T \)
97 \( 1 + (0.529 + 0.848i)T \)
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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

−23.568753836973843029000353892367, −22.5223739210745301967922839201, −21.54759421750078424511693256199, −21.05182613868821037873420001603, −20.37140981412197069255363515726, −19.38846080773520274825742617949, −18.68156366244649438737220512547, −18.21281100978577434404437914191, −16.62989915356603847852345311388, −16.12355990967344940498810055600, −15.33790780565947020100565176884, −14.36485482120680794135489924903, −13.37043549931379832229396407790, −12.0409265428180599359869047638, −11.52264298377114046581683088722, −10.579425431496316862501740902680, −9.88445327109474949057205756370, −8.51978617428927797505128017903, −8.34286806097931093929928153073, −7.49768448476528972810308951826, −5.73737853017590138905994012011, −4.43733131795583937029187038410, −3.57273813707434834561018452849, −2.807588256782613891620218498343, −1.59306594264772392093116277036, 0.565873040592445632513062076729, 1.57051149668443117210662457457, 3.04259140200971607905168707601, 4.357807877389643567747827481390, 5.4369773774981104125712930473, 6.801195208595756317278414753296, 7.5225944147557547171315361069, 8.08391032674053445248121379667, 8.714201536421745764141366469212, 10.03234331787798419971667808272, 10.908693491469682363369030886296, 11.93566492321969550522596610917, 13.02509799159920834617983267011, 13.95043818223375629938583612736, 14.705559564386943708105174888355, 15.6008362893963305223409236514, 16.24073619985948039389841084016, 17.50595489321120920642068883945, 18.072845178458042056210637427730, 18.78811324821709539592168604117, 19.874811090194614461076859816, 20.1562784364612451180861295382, 21.02722762886806005107648898691, 22.97552440664513074307066117508, 23.494174334173424113244335508359

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