Properties

Degree 1
Conductor 167
Sign $-0.126 + 0.991i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.881 − 0.472i)2-s + (0.0567 − 0.998i)3-s + (0.553 + 0.832i)4-s + (−0.752 − 0.658i)5-s + (−0.521 + 0.853i)6-s + (−0.999 − 0.0378i)7-s + (−0.0944 − 0.995i)8-s + (−0.993 − 0.113i)9-s + (0.351 + 0.936i)10-s + (0.280 + 0.959i)11-s + (0.862 − 0.505i)12-s + (−0.243 + 0.969i)13-s + (0.862 + 0.505i)14-s + (−0.700 + 0.713i)15-s + (−0.387 + 0.922i)16-s + (−0.914 − 0.404i)17-s + ⋯
L(s,χ)  = 1  + (−0.881 − 0.472i)2-s + (0.0567 − 0.998i)3-s + (0.553 + 0.832i)4-s + (−0.752 − 0.658i)5-s + (−0.521 + 0.853i)6-s + (−0.999 − 0.0378i)7-s + (−0.0944 − 0.995i)8-s + (−0.993 − 0.113i)9-s + (0.351 + 0.936i)10-s + (0.280 + 0.959i)11-s + (0.862 − 0.505i)12-s + (−0.243 + 0.969i)13-s + (0.862 + 0.505i)14-s + (−0.700 + 0.713i)15-s + (−0.387 + 0.922i)16-s + (−0.914 − 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.126 + 0.991i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.126 + 0.991i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(167\)
\( \varepsilon \)  =  $-0.126 + 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{167} (36, \cdot )$
Sato-Tate  :  $\mu(83)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 167,\ (0:\ ),\ -0.126 + 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.0004117574104 + 0.0004674776556i$
$L(\frac12,\chi)$  $\approx$  $0.0004117574104 + 0.0004674776556i$
$L(\chi,1)$  $\approx$  0.3624519511 - 0.2038647723i
$L(1,\chi)$  $\approx$  0.3624519511 - 0.2038647723i

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

−27.17691728395274147330919446528, −26.58093059875050555464727323502, −25.76680515937763423299446981491, −24.80933342771373730348207745925, −23.38598602771342240026732582800, −22.64053687293158694415691886486, −21.63489474523454786198452702631, −20.151015310338692711550696767584, −19.50325898649230634923089689016, −18.70033884733016942873231535946, −17.25767697293415706542503005581, −16.45389729054425168956254850983, −15.393739076226022126369579829547, −15.119929228388983700830963166226, −13.66821763384242917335749272111, −11.79886149834260339860499737747, −10.75811707702839070365953564922, −10.10150521919731205958659894096, −8.90333338703302705957885022822, −7.99759524246565583729838120052, −6.59057945915278392767616390519, −5.64235808728334902884932984609, −3.85308622688293935760982758203, −2.78463873723465071041790675706, −0.00064827509856262624132686211, 1.65814032519778543863623537951, 2.968200465243907510355448711904, 4.488998140527618762056540841746, 6.81989337776788033961448597273, 7.05833420571771884985854809760, 8.6804445828908348682291310586, 9.14708917422011044597476243115, 10.74768399217083945021450884077, 12.031857597195408827325869966829, 12.45751605790500124605665602739, 13.47316471061627891456126917234, 15.2296298946889315412503320944, 16.3851742856045250192987136426, 17.16083626749517445115916165247, 18.24736669544974805764241132346, 19.314051751243624371535424010335, 19.74073470110129771094555249075, 20.55771491199055762467650509938, 22.09150536735968456348419673125, 23.17526070374513310091840650462, 24.205342737284076635668216187672, 25.15469169541356229682558352301, 25.96024018837801553977997641576, 26.944888250075364131656007147634, 28.225825774581730019743070618849

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