Properties

Degree $1$
Conductor $121$
Sign $0.704 - 0.709i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.774 − 0.633i)2-s + (−0.809 + 0.587i)3-s + (0.198 − 0.980i)4-s + (0.897 − 0.441i)5-s + (−0.254 + 0.967i)6-s + (0.0855 + 0.996i)7-s + (−0.466 − 0.884i)8-s + (0.309 − 0.951i)9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.993 − 0.113i)13-s + (0.696 + 0.717i)14-s + (−0.466 + 0.884i)15-s + (−0.921 − 0.389i)16-s + (−0.564 − 0.825i)17-s + (−0.362 − 0.931i)18-s + ⋯
L(s,χ)  = 1  + (0.774 − 0.633i)2-s + (−0.809 + 0.587i)3-s + (0.198 − 0.980i)4-s + (0.897 − 0.441i)5-s + (−0.254 + 0.967i)6-s + (0.0855 + 0.996i)7-s + (−0.466 − 0.884i)8-s + (0.309 − 0.951i)9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.993 − 0.113i)13-s + (0.696 + 0.717i)14-s + (−0.466 + 0.884i)15-s + (−0.921 − 0.389i)16-s + (−0.564 − 0.825i)17-s + (−0.362 − 0.931i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.704 - 0.709i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.704 - 0.709i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.704 - 0.709i$
Motivic weight: \(0\)
Character: $\chi_{121} (47, \cdot )$
Sato-Tate group: $\mu(55)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.704 - 0.709i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.315246453 - 0.5477959042i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.315246453 - 0.5477959042i\)
\(L(\chi,1)\) \(\approx\) \(1.322514500 - 0.3811844752i\)
\(L(1,\chi)\) \(\approx\) \(1.322514500 - 0.3811844752i\)

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

−29.37311183022554523984620398540, −28.53181668953129789900688314564, −26.84395278367936676850436496172, −25.96766370064271635875512444822, −24.92755608738725815152431759439, −24.00248515048730469352856763928, −23.137357812354725955626677364502, −22.34734094483269321588103667363, −21.401706962360707235928277967728, −20.22506677377429823240295533447, −18.46989247547321646319447203158, −17.59613531288428586991641929531, −16.845678951992877579441224915604, −15.79870191928181104365018998912, −14.13741293884266665395092940491, −13.61587913770643336313576918315, −12.61089057944487460711035214429, −11.25832492316479219381580933810, −10.30954623875358098764113391698, −8.2917687894527642691808985949, −6.96782706507911966235467289357, −6.29962290602228492216796749350, −5.19689563422525395582917640247, −3.72610793505749124039679817291, −1.83085396695526153646578121560, 1.501252191987432170452110401905, 3.150179676834616291586357092787, 4.78791382717571965401016870530, 5.552607947491287380599646367974, 6.440975112346704170573427777089, 9.035727085818809282015090597293, 9.807630234155885484734926272285, 11.094571702667158243967762511705, 11.93031234850330524787432074090, 13.0173742955475690431212830133, 14.08955045891217131320060626992, 15.50067567183418502161237592565, 16.157273826694939143287595518343, 17.82492908183054408937213009012, 18.42572547822875411281876027669, 20.21943383238047898458620519900, 20.99997781525541837551277532107, 21.90357377326259479604700169593, 22.421572590585700817802095475727, 23.72058878942526710436900688454, 24.62039345102379341208162081790, 25.74105953892364689934848914721, 27.44621415292873258099843940304, 28.22288247879766717354905702486, 28.895216992246496017693909095385

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