Properties

Label 1-547-547.127-r0-0-0
Degree $1$
Conductor $547$
Sign $0.909 + 0.416i$
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  + (0.343 − 0.939i)2-s + (−0.222 − 0.974i)3-s + (−0.763 − 0.645i)4-s + (0.00575 + 0.999i)5-s + (−0.991 − 0.126i)6-s + (0.999 + 0.0230i)7-s + (−0.868 + 0.495i)8-s + (−0.900 + 0.433i)9-s + (0.940 + 0.338i)10-s + (−0.845 + 0.534i)11-s + (−0.459 + 0.888i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.973 − 0.228i)15-s + (0.166 + 0.986i)16-s + (−0.397 − 0.917i)17-s + ⋯
L(s)  = 1  + (0.343 − 0.939i)2-s + (−0.222 − 0.974i)3-s + (−0.763 − 0.645i)4-s + (0.00575 + 0.999i)5-s + (−0.991 − 0.126i)6-s + (0.999 + 0.0230i)7-s + (−0.868 + 0.495i)8-s + (−0.900 + 0.433i)9-s + (0.940 + 0.338i)10-s + (−0.845 + 0.534i)11-s + (−0.459 + 0.888i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.973 − 0.228i)15-s + (0.166 + 0.986i)16-s + (−0.397 − 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.909 + 0.416i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.909 + 0.416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7004205630 + 0.1526304122i\)
\(L(\frac12)\) \(\approx\) \(0.7004205630 + 0.1526304122i\)
\(L(1)\) \(\approx\) \(0.7914586849 - 0.3617202950i\)
\(L(1)\) \(\approx\) \(0.7914586849 - 0.3617202950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.343 - 0.939i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.00575 + 0.999i)T \)
7 \( 1 + (0.999 + 0.0230i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.397 - 0.917i)T \)
19 \( 1 + (-0.577 + 0.816i)T \)
23 \( 1 + (0.740 + 0.671i)T \)
29 \( 1 + (-0.0862 + 0.996i)T \)
31 \( 1 + (0.449 + 0.893i)T \)
37 \( 1 + (-0.958 + 0.283i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.586 - 0.809i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (-0.684 - 0.729i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (-0.717 + 0.696i)T \)
67 \( 1 + (0.998 - 0.0460i)T \)
71 \( 1 + (-0.311 + 0.950i)T \)
73 \( 1 + (0.973 + 0.228i)T \)
79 \( 1 + (-0.994 - 0.103i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (0.962 - 0.272i)T \)
97 \( 1 + (-0.109 - 0.994i)T \)
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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

−23.47494123998580758642240943719, −22.54382308530640294914025620488, −21.44217850819074201110164800039, −21.2604874611090028765846982662, −20.327903236989990866980201405806, −19.105515894571428218990106894, −17.69509537614325098676160209952, −17.17381841411483370490586426102, −16.6605641776599262949045198474, −15.44797986701220362666055694395, −15.2308498712464698481951069575, −14.14581285908121688939092263310, −13.19685274351365283902977742886, −12.31251137361981327590792519862, −11.282456353741794765627192200128, −10.24602718112374306867957521219, −9.086495999468681293980698590134, −8.458982386384786855619166079165, −7.71134192906216113696091929606, −6.21840446498671638869443066847, −5.23630879450871520910608135567, −4.76634403649708341345652526296, −4.00120874725583833070145368556, −2.50202175135308518208089659072, −0.3333707887609542196343879750, 1.55087150176700134683766534876, 2.329940260610638825233639023914, 3.15981304236502211591653117061, 4.77890289844518849259997172736, 5.391856811030101984171358390263, 6.74672737526850295943470204255, 7.53041980377288408383057081824, 8.5402867546864891218831528907, 9.88627131903181326292459794435, 10.76834650321658132414055514055, 11.43388154668879913044989705345, 12.20958057458782130239522720628, 13.03498460977791616432811258906, 14.12122278285226434630834530131, 14.46303596139332174283961714312, 15.473726679527711050242318553261, 17.26701779074309128638480085076, 17.82983927317770786420568368076, 18.51814048476321742238966870140, 19.135975557367526961892648130854, 20.053309079852029714793070146519, 20.944914788883630726125419303724, 21.79202139773634820382501034883, 22.68934369596202034457116554567, 23.28528280215549812395494601179

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