Properties

Degree 1
Conductor 61
Sign $0.595 + 0.803i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (0.743 + 0.669i)6-s + (0.406 − 0.913i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.743 + 0.669i)10-s + i·11-s + (0.669 + 0.743i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (0.207 − 0.978i)17-s + ⋯
L(s,χ)  = 1  + (0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (0.743 + 0.669i)6-s + (0.406 − 0.913i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.743 + 0.669i)10-s + i·11-s + (0.669 + 0.743i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (0.207 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.595 + 0.803i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.595 + 0.803i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.595 + 0.803i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (2, \cdot )$
Sato-Tate  :  $\mu(60)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 61,\ (1:\ ),\ 0.595 + 0.803i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(3.056810096 + 1.539653966i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(3.056810096 + 1.539653966i\)
\(L(\chi,1)\)  \(\approx\)  \(2.162905402 + 0.7015393805i\)
\(L(1,\chi)\)  \(\approx\)  \(2.162905402 + 0.7015393805i\)

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

−31.65287617018018287103706001491, −31.387139625133679555110025081650, −30.1979964111145380945478400785, −29.06261157747973655526016901201, −27.87250286391893554722090148855, −26.25989708780992645354555022883, −24.88260380311178063627568915209, −24.236657236245812379972182186409, −23.48389846659874297689908343806, −21.62031426184870613042389800336, −20.97545071142486651160250813466, −19.51298577484744484874200081533, −19.00648500903626035364585623565, −16.85231316539044042165131792240, −15.4238360577976676769677004158, −14.61536802298279399685817046725, −13.25577417084038043208536545165, −12.32893324401736749264064589912, −11.31780702166872876133436341775, −8.98564405574397241905095369722, −7.91699955261102630072061639319, −6.32731732881859186548327858266, −4.70142537379903443987494111653, −3.21825284006014094219871293014, −1.6392777144862769547634036168, 2.50492478939798634306244978568, 3.82008907179530277404721430885, 4.84182423162079509317457083765, 7.105127777473279405411257280035, 7.86375865433077037807682358532, 10.127405964020545874089439518721, 11.07758745729257320840860973556, 12.68448383366886363246489862068, 14.09967608395336902810296034606, 14.86245349736555501342009724897, 15.70454318366295968545515403439, 17.18081477538689481751735778074, 19.2026147286988288693072449225, 20.27648856484270043450312228492, 20.96085046906978597916947012826, 22.54969297999003630692772723910, 23.07240337086087907896122197701, 24.60022195064083813877019340647, 25.6718958272306965271604174096, 26.72489355221196817206648440708, 27.719064875578742372841281421700, 29.73056870068078452650451468793, 30.44809158626540686023945285589, 31.35557117768247956236681792947, 32.28492932795276832255978359097

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