Properties

Degree 1
Conductor 41
Sign $0.297 - 0.954i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.809 − 0.587i)2-s i·3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.587 − 0.809i)6-s + (−0.587 + 0.809i)7-s + (−0.309 − 0.951i)8-s − 9-s + (0.309 + 0.951i)10-s + (0.951 − 0.309i)11-s + (−0.951 − 0.309i)12-s + (0.587 + 0.809i)13-s + i·14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + ⋯
L(s,χ)  = 1  + (0.809 − 0.587i)2-s i·3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.587 − 0.809i)6-s + (−0.587 + 0.809i)7-s + (−0.309 − 0.951i)8-s − 9-s + (0.309 + 0.951i)10-s + (0.951 − 0.309i)11-s + (−0.951 − 0.309i)12-s + (0.587 + 0.809i)13-s + i·14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.297 - 0.954i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.297 - 0.954i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.297 - 0.954i$
motivic weight  =  \(0\)
character  :  $\chi_{41} (2, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 41,\ (0:\ ),\ 0.297 - 0.954i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8928840714 - 0.6570267233i$
$L(\frac12,\chi)$  $\approx$  $0.8928840714 - 0.6570267233i$
$L(\chi,1)$  $\approx$  1.154853410 - 0.5989474567i
$L(1,\chi)$  $\approx$  1.154853410 - 0.5989474567i

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

−35.139605886665011891542894174165, −33.46746468988139399551548205095, −32.782264608753106317951239389613, −32.06960522527543339386207832608, −30.87250889760960142097538974598, −29.372629557053296332903476055077, −27.89296849863771461471009541777, −26.74855378775485749123975512401, −25.56721072540963115339447807669, −24.36626610625176020449402040024, −22.92509956413567631018660006860, −22.26809236999670496068729801821, −20.50569673847266237240268839597, −20.189197204386315098107499683730, −17.30219971276130379306458995832, −16.40290194090604759291620231081, −15.55211626616948348757573474136, −14.10619413649570521421724221429, −12.79505015039196794519074475876, −11.339291492537046572840576796046, −9.49656832298830935138564509652, −8.03058837244419362719445192743, −6.11492251627196633975151480423, −4.543044347580638406677220510781, −3.61235238636611322277255481423, 2.12833102826280710218526300898, 3.57366127330464903853511056556, 6.01964263880981140984671889665, 6.87482551716963960156762497899, 9.135822946699873899647360707310, 11.19737683867800183305950448029, 11.94546368102915107242660114156, 13.39370160693107082569299162048, 14.41472864435699897739947345948, 15.77500181988167621211055406445, 18.04150959896894170148699016939, 19.118391028977642381192585469213, 19.77446348792423345635446136098, 21.80822144904119064614744579094, 22.55248036125406956426520097660, 23.75790317084382506359138361710, 24.83363566024090379994327202211, 26.16158071979735830484079850639, 28.1057073592349070702758750373, 29.090393401725065972443071247369, 30.241002661010403263797446384058, 30.89673516830975951010523076133, 31.96990005380781094914519488794, 33.50071590821009769511061212157, 34.743805665274508346415135787993

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