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Properties

Label 1-547-547.441-r0-0-0
Degree $1$
Conductor $547$
Sign $0.297 - 0.954i$
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  + (0.948 − 0.316i)2-s + 3-s + (0.799 − 0.600i)4-s + (0.428 − 0.903i)5-s + (0.948 − 0.316i)6-s + (−0.200 − 0.979i)7-s + (0.568 − 0.822i)8-s + 9-s + (0.120 − 0.992i)10-s + (−0.845 + 0.534i)11-s + (0.799 − 0.600i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.428 − 0.903i)15-s + (0.278 − 0.960i)16-s + (−0.0402 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)2-s + 3-s + (0.799 − 0.600i)4-s + (0.428 − 0.903i)5-s + (0.948 − 0.316i)6-s + (−0.200 − 0.979i)7-s + (0.568 − 0.822i)8-s + 9-s + (0.120 − 0.992i)10-s + (−0.845 + 0.534i)11-s + (0.799 − 0.600i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.428 − 0.903i)15-s + (0.278 − 0.960i)16-s + (−0.0402 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.297 - 0.954i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.297 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.878883110 - 2.118540539i\)
\(L(\frac12)\) \(\approx\) \(2.878883110 - 2.118540539i\)
\(L(1)\) \(\approx\) \(2.297179398 - 1.003197978i\)
\(L(1)\) \(\approx\) \(2.297179398 - 1.003197978i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.948 - 0.316i)T \)
3 \( 1 + T \)
5 \( 1 + (0.428 - 0.903i)T \)
7 \( 1 + (-0.200 - 0.979i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.0402 + 0.999i)T \)
19 \( 1 + (0.278 + 0.960i)T \)
23 \( 1 + (0.987 - 0.160i)T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.970 + 0.239i)T \)
37 \( 1 + (0.987 + 0.160i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (-0.632 + 0.774i)T \)
53 \( 1 + (-0.996 + 0.0804i)T \)
59 \( 1 + (0.278 - 0.960i)T \)
61 \( 1 + (0.948 - 0.316i)T \)
67 \( 1 + (-0.919 - 0.391i)T \)
71 \( 1 + (0.692 - 0.721i)T \)
73 \( 1 + (0.428 + 0.903i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.845 - 0.534i)T \)
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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

−23.612696601013166284543493485688, −22.4540297233248820070048176470, −21.92651303625093585768620651609, −21.265798346402259606336464999552, −20.399264957678044741756699329423, −19.49646476472716661219518103096, −18.48877564420837976011419948135, −17.90120140126685629527632589399, −16.39921151836532297744950407619, −15.49806428694767576970145387112, −15.019556375478503069789403404386, −14.30566650573779048591661428691, −13.24185433263927790494667123734, −12.960653843458068606314133064880, −11.60117019906642513116100795228, −10.70216002127695176950028323655, −9.58906031891726629783834209461, −8.59613491784703281052961308387, −7.487371883851372171611554557667, −6.90843950980964087053514583346, −5.6178450613556974322567257403, −4.9655287189201415590937972297, −3.20590236114282284551966970220, −2.95908737827872229845805703309, −2.10456713101372411992329301289, 1.39530981347840624389273638788, 2.13287701160849997512894108491, 3.41231384294031196143681702985, 4.31171220727871395347999219731, 4.9900648220762616729771320156, 6.338584176754843485295417737, 7.36141465700651070126035418933, 8.21088253420529673038114268716, 9.677544901792039827490794666607, 9.975587137185276445414913100333, 11.188813647455082157824440621669, 12.68467810346100918455446847642, 12.86004778351432536813333727859, 13.77089322933163905375599827546, 14.48551437165262215647816537626, 15.32372127379491967810317374501, 16.37321330798420484258364374558, 16.979679776344222708039749093259, 18.48043540788224325097802950582, 19.45125399455141310119071179174, 20.08005514760046171061788169154, 20.84408682112945114706041087299, 21.20039140618840402344154623161, 22.2231671728669777656325205731, 23.58394636345131736292982246374

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