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Properties

Label 1-547-547.160-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.336 + 0.941i$
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  + (−0.519 + 0.854i)2-s + (−0.900 − 0.433i)3-s + (−0.459 − 0.888i)4-s + (−0.439 + 0.898i)5-s + (0.838 − 0.544i)6-s + (−0.244 − 0.969i)7-s + (0.997 + 0.0689i)8-s + (0.623 + 0.781i)9-s + (−0.539 − 0.842i)10-s + (−0.845 − 0.534i)11-s + (0.0287 + 0.999i)12-s + (−0.733 + 0.680i)13-s + (0.955 + 0.294i)14-s + (0.785 − 0.618i)15-s + (−0.577 + 0.816i)16-s + (0.756 − 0.654i)17-s + ⋯
L(s)  = 1  + (−0.519 + 0.854i)2-s + (−0.900 − 0.433i)3-s + (−0.459 − 0.888i)4-s + (−0.439 + 0.898i)5-s + (0.838 − 0.544i)6-s + (−0.244 − 0.969i)7-s + (0.997 + 0.0689i)8-s + (0.623 + 0.781i)9-s + (−0.539 − 0.842i)10-s + (−0.845 − 0.534i)11-s + (0.0287 + 0.999i)12-s + (−0.733 + 0.680i)13-s + (0.955 + 0.294i)14-s + (0.785 − 0.618i)15-s + (−0.577 + 0.816i)16-s + (0.756 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2005223372 + 0.2846441484i\)
\(L(\frac12)\) \(\approx\) \(0.2005223372 + 0.2846441484i\)
\(L(1)\) \(\approx\) \(0.4409772056 + 0.1411803683i\)
\(L(1)\) \(\approx\) \(0.4409772056 + 0.1411803683i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.519 + 0.854i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.439 + 0.898i)T \)
7 \( 1 + (-0.244 - 0.969i)T \)
11 \( 1 + (-0.845 - 0.534i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (0.756 - 0.654i)T \)
19 \( 1 + (-0.998 - 0.0575i)T \)
23 \( 1 + (-0.0632 - 0.997i)T \)
29 \( 1 + (0.509 + 0.860i)T \)
31 \( 1 + (0.770 + 0.636i)T \)
37 \( 1 + (0.740 + 0.671i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.658 - 0.752i)T \)
47 \( 1 + (-0.632 - 0.774i)T \)
53 \( 1 + (0.143 + 0.989i)T \)
59 \( 1 + (0.278 + 0.960i)T \)
61 \( 1 + (0.343 + 0.939i)T \)
67 \( 1 + (-0.880 - 0.474i)T \)
71 \( 1 + (-0.131 + 0.991i)T \)
73 \( 1 + (0.785 + 0.618i)T \)
79 \( 1 + (0.322 + 0.946i)T \)
83 \( 1 + (-0.996 - 0.0804i)T \)
89 \( 1 + (-0.985 + 0.171i)T \)
97 \( 1 + (0.709 + 0.705i)T \)
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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

−23.01378519186849703013642071495, −22.13028152132488089173894297778, −21.15256603899601756859585383524, −20.98138421335054680952196518054, −19.65224717507385378430875679747, −19.07998108343965024963094469679, −17.949827451279698281343448363639, −17.378257628712700667868926332879, −16.54964858221698431966663277907, −15.685122680467773671230911199458, −14.98739378679169508509091234809, −12.97811113299699157263014303535, −12.67606937644501905125182727870, −11.92587995579174605928232140525, −11.14519547951843439240435408737, −9.93695037588129403692160520822, −9.62224399067831170340260523541, −8.32408950232753174183373911424, −7.6786944857357446905920312380, −6.02509235220911190478420723721, −5.05248856938561258998182517021, −4.325952750386697085241785805201, −3.08787082268315409282056063937, −1.81196578798637031128743631978, −0.34116188129595701314237870961, 0.86873787381563147228963981079, 2.532893632471697438216806232176, 4.18962369523818666591553546771, 5.09649238941319671962544971909, 6.282095240370497956976793818239, 6.96112462895228695354349557554, 7.50174976543619431238729756241, 8.49712604012295026087649989853, 10.22070848878883856157060745837, 10.359345067404170394242109754099, 11.362464374257823842190855497687, 12.49599487074199074925276859182, 13.68421377166295540900959889807, 14.26039058124387218865903356589, 15.36942652514325501113618783637, 16.35085446595435530694414902031, 16.75131055100340411402381555646, 17.71852478259226560525545016935, 18.58697224661508905731407526432, 19.02453645920942714743783892938, 19.86992706192348960302840741011, 21.35739176446435818653982700275, 22.40741362478998491075064018321, 23.05077058752314294908786749000, 23.70057130009886110285856918134

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