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Properties

Label 1-547-547.295-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.701 - 0.712i$
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  + (−0.0862 + 0.996i)2-s + (0.623 + 0.781i)3-s + (−0.985 − 0.171i)4-s + (0.256 + 0.966i)5-s + (−0.832 + 0.553i)6-s + (0.509 + 0.860i)7-s + (0.256 − 0.966i)8-s + (−0.222 + 0.974i)9-s + (−0.985 + 0.171i)10-s + (−0.970 + 0.239i)11-s + (−0.479 − 0.877i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.596 + 0.802i)15-s + (0.940 + 0.338i)16-s + (0.449 − 0.893i)17-s + ⋯
L(s)  = 1  + (−0.0862 + 0.996i)2-s + (0.623 + 0.781i)3-s + (−0.985 − 0.171i)4-s + (0.256 + 0.966i)5-s + (−0.832 + 0.553i)6-s + (0.509 + 0.860i)7-s + (0.256 − 0.966i)8-s + (−0.222 + 0.974i)9-s + (−0.985 + 0.171i)10-s + (−0.970 + 0.239i)11-s + (−0.479 − 0.877i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.596 + 0.802i)15-s + (0.940 + 0.338i)16-s + (0.449 − 0.893i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4663699679 + 1.113301503i\)
\(L(\frac12)\) \(\approx\) \(-0.4663699679 + 1.113301503i\)
\(L(1)\) \(\approx\) \(0.4774953139 + 0.9458942188i\)
\(L(1)\) \(\approx\) \(0.4774953139 + 0.9458942188i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.0862 + 0.996i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.256 + 0.966i)T \)
7 \( 1 + (0.509 + 0.860i)T \)
11 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.449 - 0.893i)T \)
19 \( 1 + (-0.539 - 0.842i)T \)
23 \( 1 + (-0.154 + 0.987i)T \)
29 \( 1 + (0.675 - 0.736i)T \)
31 \( 1 + (0.851 - 0.524i)T \)
37 \( 1 + (-0.928 + 0.370i)T \)
41 \( 1 + T \)
43 \( 1 + (0.0517 + 0.998i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (-0.596 + 0.802i)T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (0.990 - 0.137i)T \)
67 \( 1 + (-0.479 - 0.877i)T \)
71 \( 1 + (-0.994 - 0.103i)T \)
73 \( 1 + (-0.596 - 0.802i)T \)
79 \( 1 + (0.0517 + 0.998i)T \)
83 \( 1 + (0.885 - 0.464i)T \)
89 \( 1 + (0.990 + 0.137i)T \)
97 \( 1 + (0.978 - 0.205i)T \)
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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

−23.0454820089155134593317330430, −21.635650874754741033168824788621, −20.80518376017269492638030918523, −20.496988984962896161296754870388, −19.55397619551800795993262041313, −18.964982901765688325514210985813, −17.77641490740161340132191670740, −17.39254441406385101298955520545, −16.36501523512384090225759263228, −14.75351629922914260430317401647, −14.07917415259743673866982111793, −13.23032989644759838311364278713, −12.56078184963667924356707304024, −12.03593365536211204234311671834, −10.48171694333406023762341344459, −10.10267119063415394132916278812, −8.61884135402198499781063484959, −8.295964598760230371033048321, −7.35937306766327415667621614395, −5.738897115726238723159992552, −4.72251952459838288444942822027, −3.73324028954788022552305852018, −2.517888858583985528792855175328, −1.61209585241887200169923049612, −0.59761455315241782184329655303, 2.26564642807306701912516688038, 3.0235637139525581360762336935, 4.52859875740511381491157225548, 5.14875504597982123697650634116, 6.18936129454938337637141095723, 7.467425999614587081152045030450, 7.96495000360828345255590271360, 9.21299012139797386443211527508, 9.73069465730264397158246786255, 10.670058077353635050640887637363, 11.81456731372031536254418791172, 13.252877782196073996383923185053, 14.0897190031425013578605559907, 14.68332437088272510162424836282, 15.5243473574207946556640569122, 15.823303522481160916583530239705, 17.258121176583925547765652327680, 17.83665988557080186326608126599, 18.913048430952744841944081577633, 19.32481656332163819667029732559, 20.91369757615601929356765225237, 21.54231618223669810165529752507, 22.202576616889035698887918869368, 23.014183662677152165710669191813, 24.04714359714788768997982759455

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