Properties

Degree 1
Conductor 311
Sign $0.851 + 0.523i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.918 − 0.394i)2-s + (0.918 + 0.394i)3-s + (0.688 − 0.724i)4-s + (−0.994 − 0.101i)5-s + 6-s + (0.151 + 0.988i)7-s + (0.347 − 0.937i)8-s + (0.688 + 0.724i)9-s + (−0.954 + 0.299i)10-s + (−0.979 + 0.201i)11-s + (0.918 − 0.394i)12-s + (0.688 + 0.724i)13-s + (0.528 + 0.848i)14-s + (−0.874 − 0.485i)15-s + (−0.0506 − 0.998i)16-s + (0.954 − 0.299i)17-s + ⋯
L(s,χ)  = 1  + (0.918 − 0.394i)2-s + (0.918 + 0.394i)3-s + (0.688 − 0.724i)4-s + (−0.994 − 0.101i)5-s + 6-s + (0.151 + 0.988i)7-s + (0.347 − 0.937i)8-s + (0.688 + 0.724i)9-s + (−0.954 + 0.299i)10-s + (−0.979 + 0.201i)11-s + (0.918 − 0.394i)12-s + (0.688 + 0.724i)13-s + (0.528 + 0.848i)14-s + (−0.874 − 0.485i)15-s + (−0.0506 − 0.998i)16-s + (0.954 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.851 + 0.523i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.851 + 0.523i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(311\)
\( \varepsilon \)  =  $0.851 + 0.523i$
motivic weight  =  \(0\)
character  :  $\chi_{311} (304, \cdot )$
Sato-Tate  :  $\mu(62)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 311,\ (1:\ ),\ 0.851 + 0.523i)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.107998876 + 1.161637469i$
$L(\frac12,\chi)$  $\approx$  $4.107998876 + 1.161637469i$
$L(\chi,1)$  $\approx$  2.249043657 + 0.1209511655i
$L(1,\chi)$  $\approx$  2.249043657 + 0.1209511655i

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

−24.61279582249814617703041229609, −24.09562691174323215569576501920, −23.18034024145163503678305233380, −22.71882804633081716927310183737, −20.98808476522811949180385877526, −20.665087499238668007038297538680, −19.80505433250665615935875495260, −18.80279519422420235939203088566, −17.713449143673656833375476657383, −16.29593077099554893207448855723, −15.72003599143966113809087366897, −14.74821803264780598172491369517, −13.97603970954332672330947594198, −13.12698215186813060796654282112, −12.361975084463879033200697608350, −11.18818671527389989952043027115, −10.17412845249798628698142565462, −8.28312788414349295359498508951, −7.84076105526027520359337973824, −7.13593364029925537845648094411, −5.78323334139190302604862523118, −4.33043714963558132359217092579, −3.56420705348617971824577538498, −2.696114221733624493402666822855, −0.88118385641538218658569431614, 1.49788198557222050519121211196, 2.838596029031531403804108852076, 3.51398586070746669723702624811, 4.69624041231699902456108226523, 5.51930012902317221395392225236, 7.17869795304844682872983244298, 8.078261933922759269744099126429, 9.20888973037775272269212109566, 10.27216470580168563798558160332, 11.4538744150055528497578114608, 12.14017620825576670566015008518, 13.22847957667626806218941129247, 14.12178220367459447118543129005, 15.069547665971303436744822179752, 15.751905536847386671860829694656, 16.217935035343960673789082520479, 18.53764669071036153732455374075, 18.87953148265970096320600179123, 19.9921566049864804444191409366, 20.702767189874749651827922359975, 21.399874378060506961796707178291, 22.26256572608714539466427140046, 23.39205773670827330565805103338, 24.02760606368154212278638059946, 25.02724668377474645991954811311

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