Properties

Degree 1
Conductor 61
Sign $-0.180 - 0.983i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.180 - 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.180 - 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(61\)
\( \varepsilon \)  =  $-0.180 - 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{61} (37, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 61,\ (1:\ ),\ -0.180 - 0.983i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9605985573 - 1.153378356i$
$L(\frac12,\chi)$  $\approx$  $0.9605985573 - 1.153378356i$
$L(\chi,1)$  $\approx$  0.9200223296 - 0.5781922535i
$L(1,\chi)$  $\approx$  0.9200223296 - 0.5781922535i

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

−32.99234239165709658783276095191, −31.416082783097034457662541227210, −31.05852231673508282283044768751, −28.58625811340366211627066236655, −27.916468624749715065414557275003, −27.04838662388097532095509397847, −25.74300202076205741852007013480, −24.94610374349597626512764235215, −24.059111791889619539898091142893, −22.56331448247042394800774784869, −20.8898896662156903189261491836, −20.11615183067527987052874936257, −18.80589084959483670305537086829, −17.55683869120718412623572777191, −16.11601436715816141859167269713, −15.435517991633923295077054651327, −14.3677449273071475913716385242, −12.876718213408038383256722991194, −10.94605750845168363650931863189, −9.29111007670792779188675820251, −8.69470352364436128045047708395, −7.54805750889493578235339741771, −5.447468518343120997306431323465, −4.31085857840227942807945091736, −1.74142487925873831305144242758, 1.02089241963732251659262439122, 2.79016874802023989415264285272, 3.88330498078853627410070230971, 6.87883703929164044981871516220, 7.91432422607200669570060720450, 9.07032534296070579973307971462, 10.76027635678723243017446626444, 11.516317920767679075835170915970, 13.376596196259625462715244510972, 14.00790773858666180632774626140, 15.72377923283646963620346143481, 17.54499034557236254004745682975, 18.45073037555951310978892274512, 19.362717146104247032870986110326, 20.35121483122237218473046267202, 21.40688330015312300555494717696, 22.863346187071786667263441728587, 24.160990264969102619051694928611, 25.63883745997412805998444966424, 26.66685714332376400845991516118, 27.09076740517131312046037647581, 28.92169828787039271890730988907, 29.93686075299808910059369538205, 30.62808570019724883861211334227, 31.34964665157505199977098645748

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