Properties

Degree 1
Conductor 167
Sign $-0.999 + 0.0151i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.351 + 0.936i)2-s + (−0.0189 + 0.999i)3-s + (−0.752 + 0.658i)4-s + (0.280 + 0.959i)5-s + (−0.942 + 0.334i)6-s + (0.489 + 0.872i)7-s + (−0.881 − 0.472i)8-s + (−0.999 − 0.0378i)9-s + (−0.800 + 0.599i)10-s + (−0.0944 − 0.995i)11-s + (−0.644 − 0.764i)12-s + (0.822 + 0.569i)13-s + (−0.644 + 0.764i)14-s + (−0.965 + 0.261i)15-s + (0.132 − 0.991i)16-s + (0.614 − 0.788i)17-s + ⋯
L(s,χ)  = 1  + (0.351 + 0.936i)2-s + (−0.0189 + 0.999i)3-s + (−0.752 + 0.658i)4-s + (0.280 + 0.959i)5-s + (−0.942 + 0.334i)6-s + (0.489 + 0.872i)7-s + (−0.881 − 0.472i)8-s + (−0.999 − 0.0378i)9-s + (−0.800 + 0.599i)10-s + (−0.0944 − 0.995i)11-s + (−0.644 − 0.764i)12-s + (0.822 + 0.569i)13-s + (−0.644 + 0.764i)14-s + (−0.965 + 0.261i)15-s + (0.132 − 0.991i)16-s + (0.614 − 0.788i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.0151i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.0151i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $-0.999 + 0.0151i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (32, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ -0.999 + 0.0151i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.009648548722 + 1.277419825i$
$L(\frac12,\chi)$  $\approx$  $0.009648548722 + 1.277419825i$
$L(\chi,1)$  $\approx$  0.6048154433 + 1.040586662i
$L(1,\chi)$  $\approx$  0.6048154433 + 1.040586662i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−27.683235487209661289331780351122, −26.130827206866995253844106961548, −24.99592636775352020348601914254, −23.96397513520665818610416458888, −23.37065689371668086080991206989, −22.46098492752146197461362817436, −20.792984886980836610445979133726, −20.476869459712018573185414097168, −19.603995992303513408881789957433, −18.26774800215941889196000614145, −17.66612433104562202129707047344, −16.56288096831015188060891722262, −14.68570255485942486404569559178, −13.84324458257046035995724257923, −12.80474262202611116304557933188, −12.402214999746848818263720195406, −11.07531781575680082918660982697, −10.01762766972636517639784312089, −8.6494573644016303203213637196, −7.67408570182160656430971297820, −6.00100206778005699229222836349, −4.94446623025748359670392694487, −3.62925847152777047119816476451, −1.86233110135706476173780055718, −1.07391759267278143759267290488, 2.84257892858405791722285163060, 3.81557192250100704565997873090, 5.39084630998898899660722564496, 5.90708797685916677882349504903, 7.40276096774404785966147781762, 8.73641578942899179851148304821, 9.487440041897319842989032682841, 11.01707799660069129670559770992, 11.826260036021136344059860802894, 13.84360908395821194338736368623, 14.142888494638729179942950473807, 15.4852311173595017538865431240, 15.83388140568542919132792932524, 17.11844230030743202094755290509, 18.14844048759658161183292177635, 18.96587214018803467095083310302, 20.91393744814956213232579862821, 21.58656068318404329249759663475, 22.22945362107606933614300161492, 23.17455486382859198219143432481, 24.29504954582902063128217300094, 25.469946010418152925876111265848, 26.07535601499699239776226423982, 27.00907795417038155008664234851, 27.68649598582612643344298776089

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