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Properties

Label 1-547-547.47-r0-0-0
Degree $1$
Conductor $547$
Sign $0.140 + 0.990i$
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  + (−0.996 + 0.0804i)2-s + 3-s + (0.987 − 0.160i)4-s + (0.278 + 0.960i)5-s + (−0.996 + 0.0804i)6-s + (0.428 + 0.903i)7-s + (−0.970 + 0.239i)8-s + 9-s + (−0.354 − 0.935i)10-s + (0.799 + 0.600i)11-s + (0.987 − 0.160i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.278 + 0.960i)15-s + (0.948 − 0.316i)16-s + (−0.919 − 0.391i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0804i)2-s + 3-s + (0.987 − 0.160i)4-s + (0.278 + 0.960i)5-s + (−0.996 + 0.0804i)6-s + (0.428 + 0.903i)7-s + (−0.970 + 0.239i)8-s + 9-s + (−0.354 − 0.935i)10-s + (0.799 + 0.600i)11-s + (0.987 − 0.160i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.278 + 0.960i)15-s + (0.948 − 0.316i)16-s + (−0.919 − 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.140 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088902049 + 0.9456761770i\)
\(L(\frac12)\) \(\approx\) \(1.088902049 + 0.9456761770i\)
\(L(1)\) \(\approx\) \(1.029767094 + 0.4066991718i\)
\(L(1)\) \(\approx\) \(1.029767094 + 0.4066991718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.996 + 0.0804i)T \)
3 \( 1 + T \)
5 \( 1 + (0.278 + 0.960i)T \)
7 \( 1 + (0.428 + 0.903i)T \)
11 \( 1 + (0.799 + 0.600i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.919 - 0.391i)T \)
19 \( 1 + (0.948 + 0.316i)T \)
23 \( 1 + (-0.0402 - 0.999i)T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.748 - 0.663i)T \)
37 \( 1 + (-0.0402 + 0.999i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.632 - 0.774i)T \)
47 \( 1 + (-0.845 - 0.534i)T \)
53 \( 1 + (0.692 - 0.721i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (-0.996 + 0.0804i)T \)
67 \( 1 + (-0.632 - 0.774i)T \)
71 \( 1 + (-0.200 - 0.979i)T \)
73 \( 1 + (0.278 - 0.960i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (0.692 - 0.721i)T \)
89 \( 1 + (0.568 - 0.822i)T \)
97 \( 1 + (0.799 - 0.600i)T \)
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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

−23.64005606289961910401515134494, −21.898380936426659599603066816959, −21.26589141596876899193038518785, −20.20768546371905900796656909597, −19.9455481606386246035307666274, −19.38144654103102325048805695991, −17.870897130549047998770489424478, −17.55207503579310509277642553837, −16.45659732743684202493665474302, −15.808018692993530565672766372672, −14.74411130514802508103662756489, −13.8012797353575320077907839424, −12.98781154430425104811837435166, −11.95660827770272894542005474942, −10.84390067535259537918326844382, −9.93524138849518584436625516286, −9.125470274011401997052415028735, −8.473856340631546770009082795222, −7.64176706320614089167477640571, −6.82835735706210380722998068211, −5.375875945937409314108038087368, −4.07199334089728216758895047708, −3.07012601839626046322722134829, −1.71520080581574784470691330785, −0.97962569809034974803735082480, 1.778144365838300406641026923635, 2.27490767826563676377838211002, 3.2826260746200324660155618429, 4.75045424659881924473919020525, 6.38607409874300290224395782928, 6.94382395864445361755665013392, 7.912845738637396454789499752097, 8.879759021963970337802140592454, 9.53627350325564900967171107667, 10.235907679285823023518599164777, 11.53070339754735276644663141137, 12.05829566499035193069989952442, 13.59823139373712899082767864447, 14.660412314837584210407581042775, 14.90427720582909800593818789357, 15.85594148721360973622761513388, 16.948424034091908531625582309184, 18.19151976694117601399325397225, 18.33100997286610087373367838529, 19.31465899042945285921599474924, 20.00960820698253760521142482345, 20.89025545245554210834064992409, 21.75498709419264392728339525880, 22.45944241309988027970346933534, 24.07423663937445777743879815171

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