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Properties

Label 1-547-547.527-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.283 + 0.958i$
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  + (0.490 − 0.871i)2-s + (−0.222 − 0.974i)3-s + (−0.519 − 0.854i)4-s + (0.529 − 0.848i)5-s + (−0.958 − 0.283i)6-s + (−0.614 − 0.788i)7-s + (−0.999 + 0.0345i)8-s + (−0.900 + 0.433i)9-s + (−0.479 − 0.877i)10-s + (0.278 + 0.960i)11-s + (−0.717 + 0.696i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.944 − 0.327i)15-s + (−0.459 + 0.888i)16-s + (−0.937 − 0.349i)17-s + ⋯
L(s)  = 1  + (0.490 − 0.871i)2-s + (−0.222 − 0.974i)3-s + (−0.519 − 0.854i)4-s + (0.529 − 0.848i)5-s + (−0.958 − 0.283i)6-s + (−0.614 − 0.788i)7-s + (−0.999 + 0.0345i)8-s + (−0.900 + 0.433i)9-s + (−0.479 − 0.877i)10-s + (0.278 + 0.960i)11-s + (−0.717 + 0.696i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.944 − 0.327i)15-s + (−0.459 + 0.888i)16-s + (−0.937 − 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.283 + 0.958i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.283 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6230493670 - 0.8341638502i\)
\(L(\frac12)\) \(\approx\) \(-0.6230493670 - 0.8341638502i\)
\(L(1)\) \(\approx\) \(0.4134733191 - 0.9397137975i\)
\(L(1)\) \(\approx\) \(0.4134733191 - 0.9397137975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.490 - 0.871i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.529 - 0.848i)T \)
7 \( 1 + (-0.614 - 0.788i)T \)
11 \( 1 + (0.278 + 0.960i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.937 - 0.349i)T \)
19 \( 1 + (0.0287 + 0.999i)T \)
23 \( 1 + (-0.684 - 0.729i)T \)
29 \( 1 + (-0.868 + 0.495i)T \)
31 \( 1 + (0.940 - 0.338i)T \)
37 \( 1 + (0.932 - 0.359i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.910 - 0.413i)T \)
47 \( 1 + (0.428 + 0.903i)T \)
53 \( 1 + (0.756 - 0.654i)T \)
59 \( 1 + (0.799 - 0.600i)T \)
61 \( 1 + (-0.819 + 0.572i)T \)
67 \( 1 + (-0.244 - 0.969i)T \)
71 \( 1 + (0.658 - 0.752i)T \)
73 \( 1 + (-0.944 + 0.327i)T \)
79 \( 1 + (0.813 - 0.582i)T \)
83 \( 1 + (-0.0402 - 0.999i)T \)
89 \( 1 + (-0.0862 + 0.996i)T \)
97 \( 1 + (0.924 - 0.381i)T \)
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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

−23.88249495870976873458151302464, −22.991788691844460565002976450291, −22.07492874835830606935271754529, −21.77041038678682188593336626204, −21.30245090485109679309854980069, −19.797530674943566326578211713501, −18.70761081427696003623467089548, −17.89876410187635330398446346900, −16.978608665878648977003833782558, −16.24005984240523510620669320625, −15.39535015674019357558183729729, −14.933768335968052800741715681948, −13.79244930636589599738649838208, −13.33956278250483762644745954190, −11.72305437152993221165073297018, −11.2822830060432333147368051545, −9.88191950437194199071963174071, −9.1439486509669090696140449814, −8.43169295700792923807102996389, −6.78667545608329604203724737533, −6.204113051210136797623407726693, −5.54070514718753503061277771560, −4.30432442718660625542404298950, −3.39387371903548743139519010281, −2.52275317956254116763542912624, 0.46561437117756840948949947726, 1.54606779922934085598557606279, 2.427683262568582015223582539208, 3.79659952568212570454799806490, 4.816441874821815722758209261020, 5.85539322492583873726288708333, 6.6014407034983876111722511395, 7.87751670601812873259410166645, 8.98016790990936017945205990019, 9.96390856747845577043324596011, 10.70182860842189110590740198824, 11.95087844592095458707193975986, 12.575884022377293283845730124613, 13.23488954925315615870502441481, 13.76669623133274945760022513620, 14.81024170418795426718641153034, 16.14061039192659256066357707180, 17.17174144734591240329218572977, 17.8619369074005159394230448274, 18.63138667248793847751451960661, 19.81387991419227971051843875404, 20.223941088373496935084486160247, 20.767822616411582020183058608123, 22.31356569361427840120180378124, 22.649658594333931547521693995

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