Properties

Degree 1
Conductor 167
Sign $0.974 - 0.223i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.243 + 0.969i)2-s + (0.726 − 0.686i)3-s + (−0.881 − 0.472i)4-s + (0.351 − 0.936i)5-s + (0.489 + 0.872i)6-s + (−0.0189 + 0.999i)7-s + (0.672 − 0.739i)8-s + (0.0567 − 0.998i)9-s + (0.822 + 0.569i)10-s + (−0.800 − 0.599i)11-s + (−0.965 + 0.261i)12-s + (0.614 + 0.788i)13-s + (−0.965 − 0.261i)14-s + (−0.387 − 0.922i)15-s + (0.553 + 0.832i)16-s + (0.206 − 0.978i)17-s + ⋯
L(s,χ)  = 1  + (−0.243 + 0.969i)2-s + (0.726 − 0.686i)3-s + (−0.881 − 0.472i)4-s + (0.351 − 0.936i)5-s + (0.489 + 0.872i)6-s + (−0.0189 + 0.999i)7-s + (0.672 − 0.739i)8-s + (0.0567 − 0.998i)9-s + (0.822 + 0.569i)10-s + (−0.800 − 0.599i)11-s + (−0.965 + 0.261i)12-s + (0.614 + 0.788i)13-s + (−0.965 − 0.261i)14-s + (−0.387 − 0.922i)15-s + (0.553 + 0.832i)16-s + (0.206 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.974 - 0.223i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.974 - 0.223i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.974 - 0.223i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (6, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ 0.974 - 0.223i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.185230064 - 0.1338911623i$
$L(\frac12,\chi)$  $\approx$  $1.185230064 - 0.1338911623i$
$L(\chi,1)$  $\approx$  1.118303358 + 0.03855576641i
$L(1,\chi)$  $\approx$  1.118303358 + 0.03855576641i

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

−27.45369074862157212579518931263, −26.81329888512912909556236891166, −25.980183391843044826332499495796, −25.37964955107740577833501019277, −23.26354644279639702419142254285, −22.727900322981827427486616572886, −21.48249169724295486415013604315, −20.86511308801902387604210798954, −19.97903867772145614083531620372, −19.0774809886621386273957505748, −18.01783869287204443327687755469, −17.07697294418355766371271479427, −15.6477059279532625565900698013, −14.47286276074489142042146179646, −13.67777419616193256287991622749, −12.74693849915272214057941792428, −10.89675837682437002921341678592, −10.44980552905471938675033227569, −9.731625737168600709825552044682, −8.26973342138765982751340360756, −7.37618004319013515882706403836, −5.32996903313677020526843991678, −3.847021350602553723440452892800, −3.15840981618850838478872584868, −1.81173303241260777324137908707, 1.134670912199138732626550862734, 2.79689972356674799330354795496, 4.736198913405796199107648168334, 5.79201642770426727945260218359, 6.909959616618554099986535798207, 8.259230763774373497513639426769, 8.838309409902746497812914786472, 9.63404385612390672078407746140, 11.67713159535919847344978590870, 13.06852689128693020935844558852, 13.52036968340302947185181247033, 14.69731019455470595719017364931, 15.79923801130774538869106870081, 16.54867254514847394494724932939, 18.02783683899457248865214300803, 18.47494779880048044724734934681, 19.51154740177923920229672982549, 20.79133646886385289857628818662, 21.69796469861149561124374845393, 23.24064121159130741050738132562, 24.165844895726264950005917305006, 24.6828690719546948439864743978, 25.54370455888701767017749890102, 26.26208714820693654217142315950, 27.45887265336983698110383483804

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