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Properties

Label 1-547-547.142-r0-0-0
Degree $1$
Conductor $547$
Sign $0.357 - 0.933i$
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  + (0.983 − 0.183i)2-s + (−0.900 + 0.433i)3-s + (0.932 − 0.359i)4-s + (0.0287 − 0.999i)5-s + (−0.806 + 0.591i)6-s + (0.993 − 0.114i)7-s + (0.851 − 0.524i)8-s + (0.623 − 0.781i)9-s + (−0.154 − 0.987i)10-s + (0.948 − 0.316i)11-s + (−0.684 + 0.729i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (0.407 + 0.913i)15-s + (0.740 − 0.671i)16-s + (−0.890 − 0.454i)17-s + ⋯
L(s)  = 1  + (0.983 − 0.183i)2-s + (−0.900 + 0.433i)3-s + (0.932 − 0.359i)4-s + (0.0287 − 0.999i)5-s + (−0.806 + 0.591i)6-s + (0.993 − 0.114i)7-s + (0.851 − 0.524i)8-s + (0.623 − 0.781i)9-s + (−0.154 − 0.987i)10-s + (0.948 − 0.316i)11-s + (−0.684 + 0.729i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (0.407 + 0.913i)15-s + (0.740 − 0.671i)16-s + (−0.890 − 0.454i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.357 - 0.933i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.357 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.780149756 - 1.224674108i\)
\(L(\frac12)\) \(\approx\) \(1.780149756 - 1.224674108i\)
\(L(1)\) \(\approx\) \(1.557218959 - 0.4918965523i\)
\(L(1)\) \(\approx\) \(1.557218959 - 0.4918965523i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.983 - 0.183i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.0287 - 0.999i)T \)
7 \( 1 + (0.993 - 0.114i)T \)
11 \( 1 + (0.948 - 0.316i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.890 - 0.454i)T \)
19 \( 1 + (-0.0632 + 0.997i)T \)
23 \( 1 + (-0.857 + 0.514i)T \)
29 \( 1 + (-0.418 - 0.908i)T \)
31 \( 1 + (0.725 + 0.688i)T \)
37 \( 1 + (-0.131 - 0.991i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.00575 - 0.999i)T \)
47 \( 1 + (0.278 + 0.960i)T \)
53 \( 1 + (0.586 - 0.809i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (0.756 + 0.654i)T \)
67 \( 1 + (0.973 + 0.228i)T \)
71 \( 1 + (-0.999 + 0.0115i)T \)
73 \( 1 + (0.407 - 0.913i)T \)
79 \( 1 + (-0.868 + 0.495i)T \)
83 \( 1 + (-0.919 - 0.391i)T \)
89 \( 1 + (0.188 + 0.982i)T \)
97 \( 1 + (-0.519 + 0.854i)T \)
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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

−23.59000352386419292376094713347, −22.569090839551050597784303250433, −21.94367785684872924952814062541, −21.67946443729112615555569782916, −20.25817748161052477903624126628, −19.398466303450854024474412925315, −18.38264420356014359173798269899, −17.40121081612016633181488593976, −17.021110060040961440213931892057, −15.713304319627249203418509360558, −14.899785521505558531167381426693, −14.21670815428360278355829704153, −13.389764684125583592499290523858, −12.22478408812558963231986921288, −11.55648742515809969733600127704, −11.08321735985789975957679042817, −10.056149305041940536805107796260, −8.37646705710482904161377065674, −7.10626510950317364508527900439, −6.82355330933197850234821462949, −5.84734730564302818878021010077, −4.72881094641713786466563526140, −4.09232140740478391356986502509, −2.42260210500971839684093877402, −1.73796073096948359760531948871, 0.960745541260762992253227779015, 2.0196100809909967345459527703, 3.827775692630937670959692843811, 4.42951575022562416775270988881, 5.30385882515573875775690638908, 5.91662675208409409818196195715, 7.12575621733402151225394712721, 8.26595103806022782936455532762, 9.58388087174091661776624255098, 10.44477744314525747518371402115, 11.60189259746315056449585471313, 11.86282081205253381020084852316, 12.78700514127018085748664267938, 13.81655926792532893913091171399, 14.710281591146616362195049687521, 15.60344368703468111333926633517, 16.38012829219424599545225604099, 17.1992277726460271203380183071, 17.816510023754648192391506948249, 19.34955930684139715752502031079, 20.26309354448776555816785350450, 20.85842994590070358030481729795, 21.64435765361991781332669077003, 22.340219226173055807632688505752, 23.13488965923119037411582595754

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