Properties

Degree 1
Conductor 167
Sign $0.845 - 0.533i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.800 + 0.599i)2-s + (0.862 + 0.505i)3-s + (0.280 − 0.959i)4-s + (−0.0944 − 0.995i)5-s + (−0.993 + 0.113i)6-s + (−0.169 − 0.985i)7-s + (0.351 + 0.936i)8-s + (0.489 + 0.872i)9-s + (0.672 + 0.739i)10-s + (−0.881 − 0.472i)11-s + (0.726 − 0.686i)12-s + (−0.316 − 0.948i)13-s + (0.726 + 0.686i)14-s + (0.421 − 0.906i)15-s + (−0.843 − 0.537i)16-s + (0.954 − 0.298i)17-s + ⋯
L(s,χ)  = 1  + (−0.800 + 0.599i)2-s + (0.862 + 0.505i)3-s + (0.280 − 0.959i)4-s + (−0.0944 − 0.995i)5-s + (−0.993 + 0.113i)6-s + (−0.169 − 0.985i)7-s + (0.351 + 0.936i)8-s + (0.489 + 0.872i)9-s + (0.672 + 0.739i)10-s + (−0.881 − 0.472i)11-s + (0.726 − 0.686i)12-s + (−0.316 − 0.948i)13-s + (0.726 + 0.686i)14-s + (0.421 − 0.906i)15-s + (−0.843 − 0.537i)16-s + (0.954 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.845 - 0.533i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.845 - 0.533i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $0.845 - 0.533i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (33, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ 0.845 - 0.533i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8622099999 - 0.2492790195i$
$L(\frac12,\chi)$  $\approx$  $0.8622099999 - 0.2492790195i$
$L(\chi,1)$  $\approx$  0.8829790533 + 0.01441640776i
$L(1,\chi)$  $\approx$  0.8829790533 + 0.01441640776i

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

−27.63458593842081993798572447109, −26.65073481617689875271069730423, −25.68043143527176624688206286002, −25.50144098405551114791741877023, −24.036281750916998295174645380679, −22.753856133432140965949349110914, −21.41622592270511273620082980239, −20.98629204483744458476251783258, −19.49124401959090216493886891020, −18.86504339339693035013784746642, −18.43490833940812537034794811374, −17.24944811894981529449726817490, −15.66401810021420656505927958956, −14.89922939055119726550667919241, −13.613350824913860117156546639379, −12.42237359654565052769535554904, −11.66322196064166863714383142611, −10.17821969075255260893349822852, −9.4438712830585134548537402484, −8.16041775905588110676340259721, −7.43066497896847063164571177068, −6.21078920998010657686441779420, −3.85694329210584536460518306269, −2.66279736107209988815049634247, −1.98118182757264541222679553712, 0.89008109720362775465696223602, 2.766368240582429707859721152434, 4.4652728443619787799594103926, 5.47079405981649065656891433104, 7.28588218788197249519645378158, 8.11478858469464136688494983209, 8.92681135459832761615478330300, 10.11924509157976937523490445622, 10.72849266786511046156732888480, 12.75627332503557185071938846551, 13.750671307162939262360177042870, 14.80119765831229319290962983614, 15.94905823895253759908614158976, 16.46059747748930676339234730786, 17.49728893278519103257857903583, 18.84686332123517661133199582490, 19.87004736747714416814923506684, 20.37528937172035353857873368433, 21.33501755877954321651770146923, 23.07716131695183237378522889296, 23.94982499117806796146789439335, 24.89274249232660270613049126665, 25.694598687249593459164624648930, 26.625815396731675837185289783136, 27.311536241071992476773907615296

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