Properties

Degree 1
Conductor 113
Sign $0.922 + 0.386i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.222 + 0.974i)2-s + (0.623 − 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 + 0.781i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)14-s + 15-s + (0.623 + 0.781i)16-s + (−0.222 + 0.974i)17-s + ⋯
L(s,χ)  = 1  + (−0.222 + 0.974i)2-s + (0.623 − 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 + 0.781i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)14-s + 15-s + (0.623 + 0.781i)16-s + (−0.222 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.922 + 0.386i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.922 + 0.386i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $0.922 + 0.386i$
motivic weight  =  \(0\)
character  :  $\chi_{113} (30, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 113,\ (0:\ ),\ 0.922 + 0.386i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.146019059 + 0.2301773068i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.146019059 + 0.2301773068i\)
\(L(\chi,1)\)  \(\approx\)  \(1.129385983 + 0.2253062363i\)
\(L(1,\chi)\)  \(\approx\)  \(1.129385983 + 0.2253062363i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.81952115141552674961546370580, −28.54444074765223726942072538020, −27.33585991982636739245432402359, −26.553269190888683716791578898730, −25.409387286050346131993221016736, −24.40561210438059206541647735924, −22.77116594744786441739383961556, −21.536259407910495063359134122312, −20.9767165190416603071408393726, −20.47276104566329568625413995909, −18.976683085687304044416564378097, −18.1512540979494162582876735109, −16.76717652662983307508154665066, −15.768692371012579449165426941084, −14.142985226853079486436105048705, −13.48441476934131753955892212147, −12.093910208289214604389366385180, −10.93652896004304942660131771092, −9.764035859795019428854227563546, −8.84920581534006468001625002287, −8.16323545282552769719837629675, −5.40199655112450521296400622743, −4.57936016243303669331474492770, −2.96456615362612651489070049950, −1.796952865094316907640273560794, 1.51694243040231516803470717446, 3.35272776301203731112872568498, 5.2376644887681547526543614958, 6.59013764985913513125447776400, 7.47447496731973865309414763287, 8.36539603342969353123152377551, 9.81374387078874905783674314180, 10.91010530862367796920441465291, 13.120288532010097976417499968927, 13.58098332473680713319780621671, 14.72269389898181619924707960963, 15.45754001020621007472173703722, 17.33667377751079706340328128083, 17.84271404262276782304178770266, 18.69626397638536980660327068231, 19.93675553456582839854713558781, 21.15832611167033868771321949869, 22.6715368055503513253939282188, 23.59040587411183785390011393200, 24.364746411744606120376996712759, 25.62596823707676598541814851466, 26.01814912588732849919381012473, 26.9469358183399804740573658372, 28.33878384362923512116495262952, 29.62840162840260844708038736492

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