Properties

Degree 1
Conductor 61
Sign $0.128 + 0.991i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.128 + 0.991i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.128 + 0.991i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $0.128 + 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (53, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (1:\ ),\ 0.128 + 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8466480530 + 0.7437543014i$
$L(\frac12,\chi)$  $\approx$  $0.8466480530 + 0.7437543014i$
$L(\chi,1)$  $\approx$  0.7864257517 + 0.3034351670i
$L(1,\chi)$  $\approx$  0.7864257517 + 0.3034351670i

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

−32.24698764988621002404706949173, −30.4536814079990765568218308133, −29.67388581454726936052257613520, −28.39862273542492431906611656321, −27.98103863143168196327873795360, −26.25687662870915789063121715559, −25.18788916450783126733107817311, −24.32162033697025696742474103925, −23.67009633521738204736615016658, −21.58919456044067829229228046604, −20.46795205027669932139205603227, −19.05632252241682839453404131368, −18.12116550942446222279179725506, −17.29345717412430153211070490256, −16.29284876923033455293092479520, −14.50753378403956427181604172391, −13.28209095798751050732962522608, −11.713339196837379849814762425381, −10.65307598452781362231188116892, −8.76888907563443764540819341970, −8.117651045018955270904927293986, −6.3416855378958896946909752582, −5.48680436971798510389747524073, −2.11694309824526000138562105110, −0.92114129091361188170945823031, 1.72643011936352240851948277252, 3.61355527197429726707790627429, 5.546696503761166676997840497398, 7.202143019437921568561556496174, 8.83905349540284493695281934035, 10.07935322930967539816651351432, 10.77251914684064633363992106386, 11.9999225673362675713475063099, 14.11494763320623576384522718010, 15.33287555036582994014520629891, 16.67312532499785719055444627751, 17.681197844767134126812190821654, 18.41607663486657084267770045620, 20.401501826294746587908902215290, 20.91781004668019655809372840631, 22.016612540525157506405415896079, 23.38938432594859230898765814311, 25.32547907454835941594892968712, 25.91759127299012251441584442316, 27.23063984692629299605267862604, 27.81475202901419104435498570215, 29.026109121576860421297482930084, 30.02834838859941485387573790242, 31.173300094538918697321314673291, 33.167474681948441125367859341308

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