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Properties

Label 1-547-547.524-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.982 - 0.188i$
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  + (0.973 + 0.228i)2-s + (−0.222 − 0.974i)3-s + (0.895 + 0.444i)4-s + (−0.937 − 0.349i)5-s + (0.00575 − 0.999i)6-s + (0.143 − 0.989i)7-s + (0.770 + 0.636i)8-s + (−0.900 + 0.433i)9-s + (−0.832 − 0.553i)10-s + (−0.919 − 0.391i)11-s + (0.233 − 0.972i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.131 + 0.991i)15-s + (0.605 + 0.795i)16-s + (−0.980 − 0.194i)17-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)2-s + (−0.222 − 0.974i)3-s + (0.895 + 0.444i)4-s + (−0.937 − 0.349i)5-s + (0.00575 − 0.999i)6-s + (0.143 − 0.989i)7-s + (0.770 + 0.636i)8-s + (−0.900 + 0.433i)9-s + (−0.832 − 0.553i)10-s + (−0.919 − 0.391i)11-s + (0.233 − 0.972i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.131 + 0.991i)15-s + (0.605 + 0.795i)16-s + (−0.980 − 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.982 - 0.188i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.982 - 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08069420119 - 0.8507589406i\)
\(L(\frac12)\) \(\approx\) \(0.08069420119 - 0.8507589406i\)
\(L(1)\) \(\approx\) \(1.006514140 - 0.4505699912i\)
\(L(1)\) \(\approx\) \(1.006514140 - 0.4505699912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.973 + 0.228i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.937 - 0.349i)T \)
7 \( 1 + (0.143 - 0.989i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.980 - 0.194i)T \)
19 \( 1 + (-0.890 + 0.454i)T \)
23 \( 1 + (-0.109 - 0.994i)T \)
29 \( 1 + (-0.596 - 0.802i)T \)
31 \( 1 + (0.813 + 0.582i)T \)
37 \( 1 + (0.529 - 0.848i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.376 + 0.926i)T \)
47 \( 1 + (-0.0402 - 0.999i)T \)
53 \( 1 + (0.924 - 0.381i)T \)
59 \( 1 + (-0.200 - 0.979i)T \)
61 \( 1 + (-0.778 - 0.627i)T \)
67 \( 1 + (-0.958 + 0.283i)T \)
71 \( 1 + (-0.717 + 0.696i)T \)
73 \( 1 + (-0.131 - 0.991i)T \)
79 \( 1 + (0.990 - 0.137i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (-0.154 - 0.987i)T \)
97 \( 1 + (-0.880 - 0.474i)T \)
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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

−23.71065646462241542640699596125, −22.661794427953783368616342364526, −22.13600578575728045576193596971, −21.51086322324738027354543235662, −20.60642431853474108440788382764, −19.80961522826121215685879203760, −19.06021556572941416121637674200, −17.87270101175094017791725239668, −16.72175331303428449433943212960, −15.664748549103981335157656722859, −15.10697391530277825492466855908, −15.00460840598993395726907317898, −13.53796507787543801451681059009, −12.408809704735011097316352864945, −11.80317660825222128694019591960, −10.96690497642050278716810308198, −10.28444745926473526126482149239, −9.11573742715799386159767440565, −7.937934324999390130440402738620, −6.841662980634322535129008873106, −5.69115271401196855867965901367, −4.858294068240514557026733089835, −4.194464014015175300980855741270, −2.99255378117377264337123413114, −2.32070204997242546450175543088, 0.29684802554472338531519208107, 1.967532263257255931412070533077, 3.02161394935512097008357520998, 4.326651556686237863949117481600, 4.9305578348792414226908344829, 6.24617397711542476601158762159, 7.07432149428903337480530207663, 7.81376955918446215226589483220, 8.43018222715975905011266350323, 10.442075705551436992074843266690, 11.22262221813499745276146450336, 11.989372642604247869153830345759, 12.89455689523610336022972646285, 13.352059158549199204770747664702, 14.38826103116564261030362700863, 15.172271733789767502544995640700, 16.37707244939735052209734895745, 16.79145267237939963968462959863, 17.80850689150944013963782625588, 19.04184374797870592866148093774, 19.783824196384846384463458552972, 20.37579285773939897274724087641, 21.36574809642497131010307911947, 22.55523292444146712859036951154, 23.21767832000403870820282071775

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