Properties

Degree 1
Conductor 311
Sign $0.111 - 0.993i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.528 + 0.848i)2-s + (0.528 − 0.848i)3-s + (−0.440 + 0.897i)4-s + (−0.250 − 0.968i)5-s + 6-s + (0.918 − 0.394i)7-s + (−0.994 + 0.101i)8-s + (−0.440 − 0.897i)9-s + (0.688 − 0.724i)10-s + (0.874 + 0.485i)11-s + (0.528 + 0.848i)12-s + (−0.440 − 0.897i)13-s + (0.820 + 0.571i)14-s + (−0.954 − 0.299i)15-s + (−0.612 − 0.790i)16-s + (−0.688 + 0.724i)17-s + ⋯
L(s,χ)  = 1  + (0.528 + 0.848i)2-s + (0.528 − 0.848i)3-s + (−0.440 + 0.897i)4-s + (−0.250 − 0.968i)5-s + 6-s + (0.918 − 0.394i)7-s + (−0.994 + 0.101i)8-s + (−0.440 − 0.897i)9-s + (0.688 − 0.724i)10-s + (0.874 + 0.485i)11-s + (0.528 + 0.848i)12-s + (−0.440 − 0.897i)13-s + (0.820 + 0.571i)14-s + (−0.954 − 0.299i)15-s + (−0.612 − 0.790i)16-s + (−0.688 + 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.111 - 0.993i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.111 - 0.993i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.111 - 0.993i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (190, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ 0.111 - 0.993i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.814562222 - 1.623139459i$
$L(\frac12,\chi)$  $\approx$  $1.814562222 - 1.623139459i$
$L(\chi,1)$  $\approx$  1.502378721 - 0.2159610677i
$L(1,\chi)$  $\approx$  1.502378721 - 0.2159610677i

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

−25.084659767122532365672221808756, −24.333993058932923970298844621702, −23.06331822133797913401918650611, −22.25165280537161460769717612554, −21.6596895065324617038145927992, −20.919617202802256176086907993297, −19.97023013925481353186495003855, −19.08104794619720501192801560245, −18.44398741619295923283326639075, −17.10774146113113773966643016353, −15.76108490859953025264680366534, −14.75522616898718508412601608154, −14.398429272013038598464271539081, −13.60816606668047272623613764522, −11.86414593301300296351256889855, −11.36425807538214775193851506413, −10.5456203296515843124003078714, −9.41962506441641744163984076888, −8.693188663252170253026021985, −7.19313108823771141510111380046, −5.76624136980669560326355283894, −4.63224084679118904730808104072, −3.7750583501833145124741732070, −2.77250880787584977947531650885, −1.73336100787848026469550272276, 0.54511246795789153135133307230, 1.9342403308134976722641325412, 3.573186446434399476930019761898, 4.60218467496045686789290768238, 5.57917818983803271255739611758, 6.99264021207980083041530901547, 7.59755580445603947319384520918, 8.610112264643099446530227437252, 9.19856254515774479253881300932, 11.25084663623300576664944726847, 12.304695672605606785476322876493, 12.96053158885630516415455363672, 13.76100150141371810244325826127, 14.8675813840692841741190122107, 15.29875922000161327714792938790, 16.928115936374096531483014510314, 17.33290307628592189676759524128, 18.16544942010678457537897130364, 19.69793857292719543414407852979, 20.25419229714586555111807275272, 21.17160423249795399273272848032, 22.39453614646495373592358439498, 23.36515926913476395901631157124, 24.29519274708312144382729785967, 24.472444408227248934936348403790

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