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Properties

Label 1-547-547.205-r0-0-0
Degree $1$
Conductor $547$
Sign $0.639 - 0.768i$
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  + (−0.999 + 0.0345i)2-s + (0.623 + 0.781i)3-s + (0.997 − 0.0689i)4-s + (−0.994 − 0.103i)5-s + (−0.650 − 0.759i)6-s + (0.915 − 0.402i)7-s + (−0.994 + 0.103i)8-s + (−0.222 + 0.974i)9-s + (0.997 + 0.0689i)10-s + (−0.748 − 0.663i)11-s + (0.675 + 0.736i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.539 − 0.842i)15-s + (0.990 − 0.137i)16-s + (−0.479 − 0.877i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0345i)2-s + (0.623 + 0.781i)3-s + (0.997 − 0.0689i)4-s + (−0.994 − 0.103i)5-s + (−0.650 − 0.759i)6-s + (0.915 − 0.402i)7-s + (−0.994 + 0.103i)8-s + (−0.222 + 0.974i)9-s + (0.997 + 0.0689i)10-s + (−0.748 − 0.663i)11-s + (0.675 + 0.736i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.539 − 0.842i)15-s + (0.990 − 0.137i)16-s + (−0.479 − 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.639 - 0.768i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.639 - 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6312138915 - 0.2957792571i\)
\(L(\frac12)\) \(\approx\) \(0.6312138915 - 0.2957792571i\)
\(L(1)\) \(\approx\) \(0.6908412911 + 0.01871569863i\)
\(L(1)\) \(\approx\) \(0.6908412911 + 0.01871569863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.999 + 0.0345i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.994 - 0.103i)T \)
7 \( 1 + (0.915 - 0.402i)T \)
11 \( 1 + (-0.748 - 0.663i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.479 - 0.877i)T \)
19 \( 1 + (-0.0862 - 0.996i)T \)
23 \( 1 + (0.770 - 0.636i)T \)
29 \( 1 + (-0.0172 + 0.999i)T \)
31 \( 1 + (0.509 - 0.860i)T \)
37 \( 1 + (0.449 - 0.893i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.289 - 0.957i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (-0.539 - 0.842i)T \)
59 \( 1 + (-0.354 - 0.935i)T \)
61 \( 1 + (0.256 + 0.966i)T \)
67 \( 1 + (0.675 + 0.736i)T \)
71 \( 1 + (-0.832 - 0.553i)T \)
73 \( 1 + (-0.539 + 0.842i)T \)
79 \( 1 + (-0.289 - 0.957i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (0.256 - 0.966i)T \)
97 \( 1 + (0.386 - 0.922i)T \)
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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

−23.73539033496482595111925192738, −23.009580734904131301428021518608, −21.41986301755556878105052359864, −20.69230756171778457856001551582, −19.89388534405307430168772083160, −19.26759383035226742993362569758, −18.54852583130859901244640416873, −17.761108191837183862560866704499, −17.12766777036018795977435451482, −15.693870732427213875354066410573, −15.06411105966933296283016502374, −14.5642146063672394737480006929, −12.914513547404717589134909687074, −12.22826722587528667034134816106, −11.49684802688371874350196367018, −10.4974377923696284797680806376, −9.42337322789061827656129270437, −8.2529098124762563740526516546, −7.941139907913124145179623243283, −7.26121827697303880915292650804, −6.10750437925775182581974076612, −4.65363371510560368370801495121, −3.18198376863511365185300048972, −2.29672677119143336100554594800, −1.27301443369072680700199504025, 0.491228351843277249399442274624, 2.27066407418924158021779672143, 3.10056108223841431144246810515, 4.44862626398321821988651469850, 5.17416993975688984365656745274, 7.0448298266500605856235092160, 7.64483348580808618265046253454, 8.52477855960883652274271673990, 9.110107116811853709653974375080, 10.269257266322295347224876435136, 11.16010092580451071974513290866, 11.4787611649995318019655020241, 12.970538969081450497201700790713, 14.30612918457497715696941250619, 14.95343472157658882476530842482, 15.81538609747046244045366729397, 16.398990157243812456136691034328, 17.26214872430973020656756905617, 18.35122391838917877064196354984, 19.21989534907378279904735716962, 19.87651767404594627640249263356, 20.60163323074464606518060306957, 21.2207914363041002756777472501, 22.22377650006163097758443343847, 23.56549591107245620960015529694

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