Properties

Label 1-547-547.317-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.998 + 0.0447i$
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  + (−0.667 + 0.744i)2-s + (0.623 + 0.781i)3-s + (−0.109 − 0.994i)4-s + (−0.910 − 0.413i)5-s + (−0.998 − 0.0575i)6-s + (−0.131 − 0.991i)7-s + (0.813 + 0.582i)8-s + (−0.222 + 0.974i)9-s + (0.915 − 0.402i)10-s + (−0.632 − 0.774i)11-s + (0.709 − 0.705i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (−0.244 − 0.969i)15-s + (−0.976 + 0.216i)16-s + (−0.376 + 0.926i)17-s + ⋯
L(s)  = 1  + (−0.667 + 0.744i)2-s + (0.623 + 0.781i)3-s + (−0.109 − 0.994i)4-s + (−0.910 − 0.413i)5-s + (−0.998 − 0.0575i)6-s + (−0.131 − 0.991i)7-s + (0.813 + 0.582i)8-s + (−0.222 + 0.974i)9-s + (0.915 − 0.402i)10-s + (−0.632 − 0.774i)11-s + (0.709 − 0.705i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (−0.244 − 0.969i)15-s + (−0.976 + 0.216i)16-s + (−0.376 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.998 + 0.0447i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.998 + 0.0447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008478075147 + 0.3786864495i\)
\(L(\frac12)\) \(\approx\) \(0.008478075147 + 0.3786864495i\)
\(L(1)\) \(\approx\) \(0.5457969893 + 0.2834644751i\)
\(L(1)\) \(\approx\) \(0.5457969893 + 0.2834644751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.667 + 0.744i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.910 - 0.413i)T \)
7 \( 1 + (-0.131 - 0.991i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (-0.376 + 0.926i)T \)
19 \( 1 + (0.00575 + 0.999i)T \)
23 \( 1 + (-0.459 + 0.888i)T \)
29 \( 1 + (-0.994 + 0.103i)T \)
31 \( 1 + (0.997 - 0.0689i)T \)
37 \( 1 + (-0.763 + 0.645i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.756 + 0.654i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (-0.717 + 0.696i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (0.874 + 0.484i)T \)
67 \( 1 + (-0.965 - 0.261i)T \)
71 \( 1 + (0.143 - 0.989i)T \)
73 \( 1 + (-0.244 + 0.969i)T \)
79 \( 1 + (0.188 - 0.982i)T \)
83 \( 1 + (0.948 - 0.316i)T \)
89 \( 1 + (-0.0172 + 0.999i)T \)
97 \( 1 + (0.233 - 0.972i)T \)
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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

−22.80628862068983400424884585767, −22.14066558099438451740309809765, −20.90674825112012207787027066777, −20.37137058682161465524835469237, −19.34728284805041547026225829412, −18.91955764504424747304639156457, −18.24128503451806015544847751407, −17.560557535010348391394058378441, −16.06882237301765735199432681392, −15.46908384380443537385283440898, −14.38738990498087026411807630415, −13.32426151929913603788959084101, −12.40463617783852458383045479147, −11.85705864890628911845048177701, −11.09030050578493650972760450006, −9.750454308754551782813191364584, −8.924818512475702642420043436377, −8.24692854272128486469573991970, −7.229082839725569830312196629483, −6.69556930549175646702654075201, −4.78239951665579243584724064856, −3.624860388547779662922611566536, −2.53985969342687987949263101493, −2.07872427641974495487554565416, −0.233825693750909530931435009183, 1.35731113658972205488052758791, 3.208770709473500307999729203899, 4.069985876067218917701415703043, 5.06876511934867524467925250606, 6.101746078255090679148253867428, 7.6941821799858039140990846417, 7.92908408936291347253921931719, 8.74727139035137250665515446940, 9.94117501794365706737628831447, 10.54021445227980675577098791745, 11.326288644245776164267211389335, 13.02675956896685057682735155763, 13.73625293334681703664668474885, 14.81506973646834817367183846462, 15.45474772960412145984995125340, 16.20796974158264822042112758600, 16.74629016648391176767323307809, 17.71302742499843063034720877590, 19.04957150734639610544580589096, 19.50110960850353562874123224193, 20.33373144537569031474578551358, 20.91815014553531849004902469255, 22.35678799240529696044153893406, 23.12419538826923853065781648365, 23.935015112464031322808552617495

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