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Properties

Label 1-547-547.129-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: $\chi_{547} (129, \cdot )$
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

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

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