Properties

Degree $1$
Conductor $311$
Sign $-0.671 + 0.740i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.250 + 0.968i)2-s + (−0.250 − 0.968i)3-s + (−0.874 − 0.485i)4-s + (−0.440 − 0.897i)5-s + 6-s + (−0.994 + 0.101i)7-s + (0.688 − 0.724i)8-s + (−0.874 + 0.485i)9-s + (0.979 − 0.201i)10-s + (−0.612 − 0.790i)11-s + (−0.250 + 0.968i)12-s + (−0.874 + 0.485i)13-s + (0.151 − 0.988i)14-s + (−0.758 + 0.651i)15-s + (0.528 + 0.848i)16-s + (0.979 − 0.201i)17-s + ⋯
L(s,χ)  = 1  + (−0.250 + 0.968i)2-s + (−0.250 − 0.968i)3-s + (−0.874 − 0.485i)4-s + (−0.440 − 0.897i)5-s + 6-s + (−0.994 + 0.101i)7-s + (0.688 − 0.724i)8-s + (−0.874 + 0.485i)9-s + (0.979 − 0.201i)10-s + (−0.612 − 0.790i)11-s + (−0.250 + 0.968i)12-s + (−0.874 + 0.485i)13-s + (0.151 − 0.988i)14-s + (−0.758 + 0.651i)15-s + (0.528 + 0.848i)16-s + (0.979 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.671 + 0.740i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.671 + 0.740i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.671 + 0.740i$
Motivic weight: \(0\)
Character: $\chi_{311} (250, \cdot )$
Sato-Tate group: $\mu(31)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ -0.671 + 0.740i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.05859619273 + 0.1322118690i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.05859619273 + 0.1322118690i\)
\(L(\chi,1)\) \(\approx\) \(0.4948738611 + 0.01709635361i\)
\(L(1,\chi)\) \(\approx\) \(0.4948738611 + 0.01709635361i\)

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

−25.43139361839598384305419117591, −23.227086364983489078095565640828, −23.0352591033243093488853285870, −22.12141599896786269327033569558, −21.42638511888344517707611145198, −20.41352922914083001085062654903, −19.51906379434059546100703738831, −18.92512435094554075483037649548, −17.61656520330826867468713082985, −17.02721219460599960827731703043, −15.6465664021472407139873540923, −15.05840459428603684875901364582, −13.83838495711342045234986198991, −12.62680006057314318492970513410, −11.80936029701271756701235006602, −10.746734230654079808424726781778, −10.05143254314664247948604266678, −9.53936777637910996886189358391, −8.08596197834175317838593591729, −6.95414815651478489129229392574, −5.40454631463691952556842081525, −4.30082970086716112305868913705, −3.238710474724018364538507545470, −2.60042207863496545903832773777, −0.11709314660068523526235831419, 1.20423712526021180076282084056, 3.12165834766601194622304188048, 4.758905967766100731591100290727, 5.69066082849005450210644513831, 6.6046126784988955554993882892, 7.62938167376506349435273031141, 8.39067049548901971312344447732, 9.33257076504445328578834700669, 10.55882112255323947067907598878, 12.18171635958429461040702173168, 12.69279610124783638188459725519, 13.63955670223734258760489895080, 14.57873536969545654235744936962, 15.93951006612802908721381853556, 16.5867630228974290824161499958, 17.102431583967106356468269180486, 18.484311786616322388614725900012, 19.085078505794741164858520320880, 19.677410112162970688638203198349, 21.146283635670742647551667321337, 22.47940532980723001162322489694, 23.23841185254602793979434726414, 23.8690039590923205250776198497, 24.714269397748894458801476858234, 25.26272484265050851815698895809

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