Properties

Degree $1$
Conductor $311$
Sign $0.00561 + 0.999i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.151 − 0.988i)2-s + (0.151 + 0.988i)3-s + (−0.954 − 0.299i)4-s + (0.347 − 0.937i)5-s + 6-s + (−0.250 + 0.968i)7-s + (−0.440 + 0.897i)8-s + (−0.954 + 0.299i)9-s + (−0.874 − 0.485i)10-s + (−0.758 + 0.651i)11-s + (0.151 − 0.988i)12-s + (−0.954 + 0.299i)13-s + (0.918 + 0.394i)14-s + (0.979 + 0.201i)15-s + (0.820 + 0.571i)16-s + (−0.874 − 0.485i)17-s + ⋯
L(s,χ)  = 1  + (0.151 − 0.988i)2-s + (0.151 + 0.988i)3-s + (−0.954 − 0.299i)4-s + (0.347 − 0.937i)5-s + 6-s + (−0.250 + 0.968i)7-s + (−0.440 + 0.897i)8-s + (−0.954 + 0.299i)9-s + (−0.874 − 0.485i)10-s + (−0.758 + 0.651i)11-s + (0.151 − 0.988i)12-s + (−0.954 + 0.299i)13-s + (0.918 + 0.394i)14-s + (0.979 + 0.201i)15-s + (0.820 + 0.571i)16-s + (−0.874 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.00561 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.00561 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.00561 + 0.999i$
Motivic weight: \(0\)
Character: $\chi_{311} (126, \cdot )$
Sato-Tate group: $\mu(31)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.00561 + 0.999i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.4356148726 + 0.4380691692i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.4356148726 + 0.4380691692i\)
\(L(\chi,1)\) \(\approx\) \(0.8081871751 + 0.01326439754i\)
\(L(1,\chi)\) \(\approx\) \(0.8081871751 + 0.01326439754i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.9715955072861616486939701022, −23.94775762046470915263137863633, −23.66181657231272225481996476178, −22.3870759586940069717060889824, −22.016462104750942163269232711334, −20.387146782804098855679647856743, −19.317064541685830243868862513768, −18.54775570351479847764906518564, −17.62901894027349372636493908349, −17.14226227137721401677179819098, −15.85363484447400265856988909005, −14.80024661978929325307785136783, −13.939304550907369738460502758854, −13.41501009413783701088085794111, −12.516319444799588579197562176792, −11.00873741729824198706321121998, −10.02788046860682233403174392759, −8.684723861687160499802071607987, −7.65162938643765617918811679885, −6.95006034286823579987484177738, −6.27736526004027543755547827860, −5.06965696551976101846687833565, −3.51266064729391180367872314122, −2.44148116052191770382184872530, −0.34769040313983960671340994199, 1.95931161396592963713548262709, 2.821733698150500463346809881467, 4.24737715783175323440325259386, 5.05188353521274245156503563170, 5.745659452937584459333804123447, 7.991389729316807080813031741116, 9.10558146821671590096200759812, 9.55883475062152686040980119341, 10.40353914594821671245481844125, 11.74674953007997213572314278301, 12.36003068603403800685493134485, 13.39723427630714427563044300298, 14.42030502589785321620422205896, 15.437010922263042675703888714468, 16.29533390318280836363680004239, 17.483366957967927066234672481567, 18.258179583305467423934665914925, 19.66169363870892567526170835104, 20.11731318376571381273121095082, 21.202171490114373105312633474216, 21.55537761852619804342343454196, 22.45690136801428069909733889656, 23.38670202760043411027585413755, 24.64856772013549514022490676646, 25.51655893666900190628428475561

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