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Properties

Label 1-547-547.155-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.903 - 0.427i$
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  + (0.740 − 0.671i)2-s + (−0.222 − 0.974i)3-s + (0.0976 − 0.995i)4-s + (0.993 + 0.114i)5-s + (−0.819 − 0.572i)6-s + (0.895 − 0.444i)7-s + (−0.596 − 0.802i)8-s + (−0.900 + 0.433i)9-s + (0.813 − 0.582i)10-s + (0.278 − 0.960i)11-s + (−0.991 + 0.126i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.109 − 0.994i)15-s + (−0.980 − 0.194i)16-s + (−0.311 + 0.950i)17-s + ⋯
L(s)  = 1  + (0.740 − 0.671i)2-s + (−0.222 − 0.974i)3-s + (0.0976 − 0.995i)4-s + (0.993 + 0.114i)5-s + (−0.819 − 0.572i)6-s + (0.895 − 0.444i)7-s + (−0.596 − 0.802i)8-s + (−0.900 + 0.433i)9-s + (0.813 − 0.582i)10-s + (0.278 − 0.960i)11-s + (−0.991 + 0.126i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.109 − 0.994i)15-s + (−0.980 − 0.194i)16-s + (−0.311 + 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.903 - 0.427i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.903 - 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4954023270 - 2.205247707i\)
\(L(\frac12)\) \(\approx\) \(0.4954023270 - 2.205247707i\)
\(L(1)\) \(\approx\) \(1.111330901 - 1.277942261i\)
\(L(1)\) \(\approx\) \(1.111330901 - 1.277942261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.740 - 0.671i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.993 + 0.114i)T \)
7 \( 1 + (0.895 - 0.444i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.311 + 0.950i)T \)
19 \( 1 + (0.968 + 0.250i)T \)
23 \( 1 + (-0.558 - 0.829i)T \)
29 \( 1 + (-0.154 - 0.987i)T \)
31 \( 1 + (-0.994 + 0.103i)T \)
37 \( 1 + (0.863 - 0.504i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.999 + 0.0230i)T \)
47 \( 1 + (0.428 - 0.903i)T \)
53 \( 1 + (-0.806 + 0.591i)T \)
59 \( 1 + (0.799 + 0.600i)T \)
61 \( 1 + (-0.958 + 0.283i)T \)
67 \( 1 + (0.605 + 0.795i)T \)
71 \( 1 + (0.998 - 0.0460i)T \)
73 \( 1 + (-0.109 + 0.994i)T \)
79 \( 1 + (-0.479 - 0.877i)T \)
83 \( 1 + (-0.0402 + 0.999i)T \)
89 \( 1 + (0.725 - 0.688i)T \)
97 \( 1 + (-0.577 + 0.816i)T \)
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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

−23.83333747066265187116239871483, −22.56407982635578365582161891935, −22.09549381451478248541493047203, −21.579017467101666748131622984955, −20.4854996686747804605486502008, −20.268100318643687949199228412950, −18.08796646837162864186051173191, −17.65874005742891833600668135161, −16.96556543123969092886158902548, −15.996480042332636489443861734795, −15.1925841841128196247340092890, −14.41797741754684349441501468378, −13.93576914807492475301990699365, −12.59584885295604987180535281422, −11.84047060207607980488460743351, −10.9351200763953032184753272327, −9.52218780866719861757905889997, −9.22903173662929132877586036804, −7.823391446656352011813721456554, −6.860447780740122248836807289316, −5.55388578960153883748636828491, −5.14373343266798682264118458348, −4.40951163857720177068636930276, −3.017985523709790639948506915544, −2.01224973718550247757864625120, 0.97439177247073509349096342141, 1.90142362845820194379152628349, 2.68693572909316362435016989598, 4.11751900140760089588341512717, 5.335810759067174738501555567641, 5.94396169952951695462889350078, 6.87700041778108101855002338430, 8.01510139573245906569686453302, 9.21868601698034856805580804163, 10.351854814229012057265888807773, 11.110101379020941013140335030738, 11.89624297768316119384215765824, 12.82150019853887999819395772113, 13.61417151407517221404082688161, 14.24984026432973025785182321050, 14.76253354518929720311466882169, 16.49564919188010003284964787247, 17.29791530844837769825847954687, 18.146692205582771173764381796956, 18.82711348846636154097123978349, 19.82114621349488320487706929552, 20.46932910707718599811750188769, 21.6210673673226348216244946506, 21.99633127515657394045667338219, 22.97344005533287139380385859168

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