Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.892 - 0.451i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.978 + 0.207i)2-s + (0.809 − 0.587i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)5-s + (−0.669 + 0.743i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 10-s + (0.5 − 0.866i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (0.913 − 0.406i)15-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + ⋯
L(s,χ)  = 1  + (−0.978 + 0.207i)2-s + (0.809 − 0.587i)3-s + (0.913 − 0.406i)4-s + (0.978 + 0.207i)5-s + (−0.669 + 0.743i)6-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 10-s + (0.5 − 0.866i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (0.913 − 0.406i)15-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (−0.104 + 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.892 - 0.451i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.892 - 0.451i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.892 - 0.451i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (41, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.892 - 0.451i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.282934536 - 0.5445597953i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.282934536 - 0.5445597953i\)
\(L(\chi,1)\)  \(\approx\)  \(1.245154143 - 0.09612927903i\)
\(L(1,\chi)\)  \(\approx\)  \(1.245154143 - 0.09612927903i\)

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

−17.61333124557762190450498701548, −17.1854281439211102198725518233, −16.49950921840054947429862056166, −16.01824282117747203161892789734, −15.24338428968700041970007566461, −14.55489740863937269829966939491, −13.70547008295395301882326814223, −13.29712358813785854019620548517, −12.68912361730461075141256111186, −11.39064263357932346541974213286, −10.9684976090514452951852743439, −10.20848506452368202965642349463, −9.7735211695128860526340137618, −9.23146867822577757777602611857, −8.529742311930160254671712627021, −7.92092096975637351237194760368, −7.205162397273959478835601394263, −6.405194419075779785023012484522, −5.71147589956902519186211018671, −4.61112685197524274140896013901, −3.88785873600120701188808781527, −3.09607776429562105279428862368, −2.49259102575156558475398944644, −1.42745881450627170503060203486, −1.08637130932919343204165161465, 0.79035506135591272335478240114, 1.62289401599940015551459109363, 2.126490510441502747240320646751, 3.00242851019496843293461114214, 3.373376344843930557832331604480, 5.03947178941877880043918415461, 5.754186714421478189363294098297, 6.36690956164512488026014707440, 6.87505986843254400024035009552, 7.77370220987380725252278485366, 8.42808141149680019975472084113, 8.99089983571625071339957070474, 9.41617933949620953900178877229, 10.21593445799367663838068643809, 10.780625971941670680917394019878, 11.792942366911506693436458316086, 12.41987280529751090384862564989, 12.98166062018473673680432523239, 14.003502972467564491182763788831, 14.427722883189936670466646096109, 14.933226588598618319165831066097, 15.93227327285111918439173418950, 16.21893190041897239169451671828, 17.207337035526585679993671573, 18.04616880602205255986851014265

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