Properties

Degree 1
Conductor 2011
Sign $0.999 + 0.0438i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.444 − 0.895i)2-s + (0.954 + 0.298i)3-s + (−0.604 − 0.796i)4-s + (−0.866 + 0.498i)5-s + (0.692 − 0.721i)6-s + (0.288 + 0.957i)7-s + (−0.982 + 0.186i)8-s + (0.821 + 0.569i)9-s + (0.0609 + 0.998i)10-s + (0.961 + 0.274i)11-s + (−0.338 − 0.940i)12-s + (−0.987 − 0.158i)13-s + (0.985 + 0.168i)14-s + (−0.976 + 0.217i)15-s + (−0.270 + 0.962i)16-s + (−0.0265 − 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.444 − 0.895i)2-s + (0.954 + 0.298i)3-s + (−0.604 − 0.796i)4-s + (−0.866 + 0.498i)5-s + (0.692 − 0.721i)6-s + (0.288 + 0.957i)7-s + (−0.982 + 0.186i)8-s + (0.821 + 0.569i)9-s + (0.0609 + 0.998i)10-s + (0.961 + 0.274i)11-s + (−0.338 − 0.940i)12-s + (−0.987 − 0.158i)13-s + (0.985 + 0.168i)14-s + (−0.976 + 0.217i)15-s + (−0.270 + 0.962i)16-s + (−0.0265 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.999 + 0.0438i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.999 + 0.0438i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $0.999 + 0.0438i$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (29, \cdot )$
Sato-Tate  :  $\mu(2010)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ 0.999 + 0.0438i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.534073169 + 0.07758589716i$
$L(\frac12,\chi)$  $\approx$  $3.534073169 + 0.07758589716i$
$L(\chi,1)$  $\approx$  1.603605145 - 0.2966777576i
$L(1,\chi)$  $\approx$  1.603605145 - 0.2966777576i

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

−19.62205410667223707106225507599, −19.399283484557894148002761604830, −18.142608788565850521909647655621, −17.43127088490870906405418428952, −16.61752088071848586964358552415, −16.19560112454027257891558541756, −15.06133157006928299783440925701, −14.68209059906454522280917956536, −14.14467360650325531279224588158, −13.11812025116351734146239243456, −12.778982902726907486066560863816, −11.87098547264986379159493680081, −11.03130102096983309935203955856, −9.67066732461827367084153583260, −9.00176519870739482930826618557, −8.37260942325661521044397294856, −7.47934005686472771020715134308, −7.21629421123850983372104713362, −6.36247423560395448767632397925, −4.93411963279046898442637517264, −4.44041212862916913237938256462, −3.61740679313289022216558441689, −3.05276259635083748217617011353, −1.476583623096061193542619158137, −0.5958785315212737585373134082, 0.8269659484682717273238314094, 2.06575253077501825708694774048, 2.67867290662944686388959039035, 3.38418097533569054519370010145, 4.23148358907644253688107594181, 4.8615647704462602772443412607, 5.85341435327202713834680078814, 7.11501823496337649900304001870, 7.78541190364785051930837901356, 8.78094623581453363352810931176, 9.45445062719486796632865834635, 9.9358818349554116624418254044, 11.0825663318881141502017480901, 11.710131237637912590462375627222, 12.23100733277406453967462609203, 13.08906002798334946902316409250, 14.103570102501736970584300117691, 14.61121794057808066808370661561, 15.1139163411929518097788716262, 15.67348426600520625209402762678, 16.80166293747773233582070305704, 17.989209073807010585245890837019, 18.78539582254140456504485851503, 19.043574281598783612197260296517, 19.921194304469788881275013832948

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