Properties

Degree $1$
Conductor $191$
Sign $-0.317 - 0.948i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.677 − 0.735i)2-s + (0.245 + 0.969i)3-s + (−0.0825 + 0.996i)4-s + (−0.0825 + 0.996i)5-s + (0.546 − 0.837i)6-s − 7-s + (0.789 − 0.614i)8-s + (−0.879 + 0.475i)9-s + (0.789 − 0.614i)10-s + (−0.945 + 0.324i)11-s + (−0.986 + 0.164i)12-s + (0.245 − 0.969i)13-s + (0.677 + 0.735i)14-s + (−0.986 + 0.164i)15-s + (−0.986 − 0.164i)16-s + (−0.401 + 0.915i)17-s + ⋯
L(s,χ)  = 1  + (−0.677 − 0.735i)2-s + (0.245 + 0.969i)3-s + (−0.0825 + 0.996i)4-s + (−0.0825 + 0.996i)5-s + (0.546 − 0.837i)6-s − 7-s + (0.789 − 0.614i)8-s + (−0.879 + 0.475i)9-s + (0.789 − 0.614i)10-s + (−0.945 + 0.324i)11-s + (−0.986 + 0.164i)12-s + (0.245 − 0.969i)13-s + (0.677 + 0.735i)14-s + (−0.986 + 0.164i)15-s + (−0.986 − 0.164i)16-s + (−0.401 + 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.317 - 0.948i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.317 - 0.948i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.317 - 0.948i$
Motivic weight: \(0\)
Character: $\chi_{191} (161, \cdot )$
Sato-Tate group: $\mu(38)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ -0.317 - 0.948i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.06837306867 - 0.09496765952i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.06837306867 - 0.09496765952i\)
\(L(\chi,1)\) \(\approx\) \(0.5504825316 + 0.1135242554i\)
\(L(1,\chi)\) \(\approx\) \(0.5504825316 + 0.1135242554i\)

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

−26.84712134968651788716808929465, −26.06823191056700263413483065127, −25.081364646580500821462044998904, −24.59191543926976030526196714934, −23.40388450893946672070516818179, −23.12622647400681593646564638397, −21.14491776676729466077668687513, −20.05343596596462397009054725395, −19.210013532574868220441286222336, −18.56240451860793081953439317716, −17.45730620447277937582195418951, −16.39873506593225829159944990936, −15.88271843504217107180504611510, −14.367964179361216185637383740838, −13.37198901138446721663951623388, −12.63424540091558996698421060182, −11.25258106906615544219102919831, −9.72459083805883564342374060317, −8.80415552066570297976181718053, −8.04623685284106865120189525484, −6.81870334610863337588899603392, −6.02848292094459268703508818497, −4.66431749780929996065660720152, −2.59825426221860802390802813926, −1.07165209189452243262238040585, 0.05644071349071664743309640791, 2.57888181454791413956437061318, 3.14697819505538280636775716084, 4.36013970262572827062183884889, 6.143525100080467206384466215556, 7.567114922085762174568464828837, 8.65108195008261749073554804444, 9.86639061689481267826022921933, 10.462644450998810779345209297323, 11.157129702832980424541314816132, 12.70800073551611198532652661935, 13.60069005715649393896140638134, 15.30695470302875523233139138951, 15.64003882442602286271884463870, 17.05331952762412050516632384204, 17.91123530508203562046756835413, 19.18448152773620672739740630260, 19.66540147789511929854823244238, 20.84277693343662853217978417412, 21.66630111991621256977064889629, 22.49494866927991304734251626868, 23.21525962051726669975237156876, 25.48340475849451017854252432366, 25.81524558031922603364915574172, 26.595177206079499814647008478574

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