Properties

Degree 1
Conductor 167
Sign $0.227 - 0.973i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.862 − 0.505i)2-s + (0.954 − 0.298i)3-s + (0.489 − 0.872i)4-s + (−0.169 − 0.985i)5-s + (0.672 − 0.739i)6-s + (−0.316 + 0.948i)7-s + (−0.0189 − 0.999i)8-s + (0.822 − 0.569i)9-s + (−0.644 − 0.764i)10-s + (0.0567 + 0.998i)11-s + (0.206 − 0.978i)12-s + (−0.965 + 0.261i)13-s + (0.206 + 0.978i)14-s + (−0.455 − 0.890i)15-s + (−0.521 − 0.853i)16-s + (−0.387 + 0.922i)17-s + ⋯
L(s,χ)  = 1  + (0.862 − 0.505i)2-s + (0.954 − 0.298i)3-s + (0.489 − 0.872i)4-s + (−0.169 − 0.985i)5-s + (0.672 − 0.739i)6-s + (−0.316 + 0.948i)7-s + (−0.0189 − 0.999i)8-s + (0.822 − 0.569i)9-s + (−0.644 − 0.764i)10-s + (0.0567 + 0.998i)11-s + (0.206 − 0.978i)12-s + (−0.965 + 0.261i)13-s + (0.206 + 0.978i)14-s + (−0.455 − 0.890i)15-s + (−0.521 − 0.853i)16-s + (−0.387 + 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.227 - 0.973i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.227 - 0.973i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.227 - 0.973i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (8, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 167,\ (0:\ ),\ 0.227 - 0.973i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.750066688 - 1.388594497i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.750066688 - 1.388594497i\)
\(L(\chi,1)\)  \(\approx\)  \(1.750585182 - 0.8998315234i\)
\(L(1,\chi)\)  \(\approx\)  \(1.750585182 - 0.8998315234i\)

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

−27.19965802037068184156528571618, −26.56828795822109583032980923382, −26.08139293827395225352419159905, −24.82301891167128356900040487171, −24.15523301589091969155664124753, −22.77804182664888530570552443977, −22.224156085442551022778808032880, −21.17868152815601601331047877308, −20.11664633519095927150086704836, −19.347364395842502563807646652960, −17.97793864172686112845390072682, −16.54604024359220428797896953958, −15.77462119895549697615756995960, −14.709258937275971931143086640449, −13.94066486023339938916097801554, −13.39243447320718388133581141565, −11.79655979669846080881321709259, −10.64359027218457155333562302050, −9.47836555521919408260484729249, −7.735756636934289247613914092265, −7.38626057096886399091572348944, −5.98417889547292930203024058148, −4.35943504842987265619089910095, −3.40766907938723563972999543122, −2.56269924024057225223837262473, 1.67318571622248860970036990101, 2.62471671013983630463774591198, 4.07465705189904700905855348303, 5.05479605447187853656236000729, 6.50035956638786039609475790464, 7.858173340327512069379422053790, 9.23200417100520617850345077575, 9.85607539950411132707809453451, 11.80910860358450971378349354811, 12.52147312230252279972454787925, 13.140275021121057218656386019513, 14.45041752112257118534460787933, 15.23760179688686391611132069474, 16.095353489354068905717797392088, 17.80881343032186759692461051369, 19.10065439449770276093661278732, 19.8438806804401476751885784146, 20.48727464309141900376237773542, 21.53842997656624089944389398045, 22.372449019127183468926813076559, 23.81606556528531853524776275998, 24.47337373866481336698695217804, 25.13005459954014144283614937297, 26.265611982835684109932571328626, 27.8053627615028349934233935266

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