Properties

Degree 1
Conductor 173
Sign $0.450 + 0.892i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.889 + 0.457i)2-s + (0.967 + 0.252i)3-s + (0.581 + 0.813i)4-s + (0.946 − 0.322i)5-s + (0.744 + 0.667i)6-s + (0.424 + 0.905i)7-s + (0.145 + 0.989i)8-s + (0.872 + 0.489i)9-s + (0.989 + 0.145i)10-s + (−0.0729 − 0.997i)11-s + (0.357 + 0.934i)12-s + (0.457 − 0.889i)13-s + (−0.0365 + 0.999i)14-s + (0.997 − 0.0729i)15-s + (−0.322 + 0.946i)16-s + (−0.994 − 0.109i)17-s + ⋯
L(s,χ)  = 1  + (0.889 + 0.457i)2-s + (0.967 + 0.252i)3-s + (0.581 + 0.813i)4-s + (0.946 − 0.322i)5-s + (0.744 + 0.667i)6-s + (0.424 + 0.905i)7-s + (0.145 + 0.989i)8-s + (0.872 + 0.489i)9-s + (0.989 + 0.145i)10-s + (−0.0729 − 0.997i)11-s + (0.357 + 0.934i)12-s + (0.457 − 0.889i)13-s + (−0.0365 + 0.999i)14-s + (0.997 − 0.0729i)15-s + (−0.322 + 0.946i)16-s + (−0.994 − 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.450 + 0.892i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.450 + 0.892i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.450 + 0.892i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (61, \cdot )$
Sato-Tate  :  $\mu(172)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (1:\ ),\ 0.450 + 0.892i)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.469613598 + 2.751667103i$
$L(\frac12,\chi)$  $\approx$  $4.469613598 + 2.751667103i$
$L(\chi,1)$  $\approx$  2.620138438 + 1.071678739i
$L(1,\chi)$  $\approx$  2.620138438 + 1.071678739i

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

−26.91343298151386273324290848454, −25.83953932525677733683843722648, −25.213901317795447292031703209136, −24.01360830272035632612574091429, −23.39686511209494110210013993472, −21.9939320611566370399310446346, −21.21063736523990671210981167094, −20.39741775781148026119933751701, −19.67618227858693974581558549195, −18.45755771476398444763453196906, −17.46989547596425348724897302132, −15.86548696231927285976502621256, −14.67960393557928402734258427334, −14.05982750391287446078191404764, −13.356533562832556262288337982607, −12.37965782063027586879588864949, −10.814886293475986049925441559313, −10.05595508391318685982557001516, −8.867302720716983979231130535238, −7.13997537866610187966576045595, −6.483614447939021933533641339034, −4.704132595809710285437965883153, −3.79335964363280798068498314075, −2.235569356129031857987029431520, −1.60648748963436738071789438646, 2.0040568365088884930362936965, 2.89397989818142465181746236083, 4.3326920780699180432859094874, 5.51254500764674242365694503439, 6.431311412476142779316405812090, 8.31652210606078266637642897444, 8.55345518716224750559390780188, 10.151429752670921481263098071457, 11.50817368263595271942940753257, 13.00478473831949471022331560005, 13.4579246310443695398433955783, 14.52239108752062901814708610664, 15.36275084733891422451515906514, 16.231728185238648196542420912091, 17.50676035441618666908057526173, 18.58565803269400511737195212261, 20.058651918508622749303943586090, 20.90419531373214615384163152065, 21.6574905822202098044545786725, 22.21405695033582554294534241964, 23.93344345726250259611697726072, 24.67081378095646335278573221079, 25.29474035992531265002867110901, 26.05897274790484131048374965434, 27.18393595793164612608873466528

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