Properties

Degree $1$
Conductor $547$
Sign $0.881 + 0.472i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.987 − 0.160i)2-s + 3-s + (0.948 − 0.316i)4-s + (−0.845 + 0.534i)5-s + (0.987 − 0.160i)6-s + (−0.632 + 0.774i)7-s + (0.885 − 0.464i)8-s + 9-s + (−0.748 + 0.663i)10-s + (0.278 + 0.960i)11-s + (0.948 − 0.316i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.845 + 0.534i)15-s + (0.799 − 0.600i)16-s + (0.692 + 0.721i)17-s + ⋯
L(s,χ)  = 1  + (0.987 − 0.160i)2-s + 3-s + (0.948 − 0.316i)4-s + (−0.845 + 0.534i)5-s + (0.987 − 0.160i)6-s + (−0.632 + 0.774i)7-s + (0.885 − 0.464i)8-s + 9-s + (−0.748 + 0.663i)10-s + (0.278 + 0.960i)11-s + (0.948 − 0.316i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.845 + 0.534i)15-s + (0.799 − 0.600i)16-s + (0.692 + 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.881 + 0.472i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.881 + 0.472i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.881 + 0.472i$
Motivic weight: \(0\)
Character: $\chi_{547} (21, \cdot )$
Sato-Tate group: $\mu(39)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.881 + 0.472i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(3.027010257 + 0.7609092633i\)
\(L(\frac12,\chi)\) \(\approx\) \(3.027010257 + 0.7609092633i\)
\(L(\chi,1)\) \(\approx\) \(2.232875384 + 0.2276730097i\)
\(L(1,\chi)\) \(\approx\) \(2.232875384 + 0.2276730097i\)

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

−23.59486671910174905398532892431, −22.4251018050196782673713317191, −21.701146991782810124128642478169, −20.681392639203725926037147831578, −20.074211184173109452457892645195, −19.49952446930996000485207194762, −18.702711291148162436842487443308, −16.89493445246961085816211813692, −16.23934362281195677426753325742, −15.746380737455590186098883641098, −14.62548740824973857488668119587, −13.81991093763815095715334307489, −13.41056958290185513761830624131, −12.220026884267741368606355096621, −11.6565024709414267307993169970, −10.36771138759334660016201753965, −9.2832796156247963543616593518, −8.24634931106625310450163038348, −7.34940120588115040429540279617, −6.75656842145784576827526471420, −5.249209556594237128089530623620, −4.186049804410975996921586495353, −3.591858307098225876269548689768, −2.7319622387233310879323180671, −1.173214224217682735203477666572, 1.739096839643202216597118896092, 2.83254899071939027763355736540, 3.43835162691268893415059394422, 4.332247744509195134609865204100, 5.57229262632571492642131632621, 6.73443942809781843272198493725, 7.54497970189947105390461821357, 8.35158962930287170760129679289, 9.9321824820907348475619381431, 10.24148477228007775926299271804, 11.92887964781896101554580772576, 12.278636732279521367163038441116, 13.12094264527696505118804427280, 14.45508816909709631361160856003, 14.658558476092710352214715528815, 15.79145174349898577642458077636, 15.88470649485484459466834029127, 17.698150900911371169514531998249, 18.92431414613940664558741718742, 19.41679173740826264367322495260, 20.118928308109537735609627218978, 20.87641615692699341509532835474, 21.97792006386328377475651139114, 22.50229369111472797061366868476, 23.29741254480813038433086304684

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