Properties

Degree 1
Conductor $ 2^{2} \cdot 5 $
Sign $0.850 + 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·3-s i·7-s − 9-s − 11-s i·13-s + i·17-s + 19-s + 21-s + i·23-s i·27-s − 29-s − 31-s i·33-s + i·37-s + 39-s + ⋯
L(s,χ)  = 1  + i·3-s i·7-s − 9-s − 11-s i·13-s + i·17-s + 19-s + 21-s + i·23-s i·27-s − 29-s − 31-s i·33-s + i·37-s + 39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.850 + 0.525i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.850 + 0.525i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.850 + 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{20} (7, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 20,\ (0:\ ),\ 0.850 + 0.525i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6153549914 + 0.1748094575i$
$L(\frac12,\chi)$  $\approx$  $0.6153549914 + 0.1748094575i$
$L(\chi,1)$  $\approx$  0.8595478356 + 0.1767421806i
$L(1,\chi)$  $\approx$  0.8595478356 + 0.1767421806i

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

−40.38883430358204640546448355975, −38.735795968667743215521103873541, −37.247801206047243723799357576921, −36.09441314324750012996248414038, −34.856507365903759726131828355361, −33.78567981564399359465343199462, −31.6515966356349751302737680839, −30.91844193420268949843068014374, −29.237289040625714209614816587000, −28.36597353127179346163501353088, −26.33239523022640549254450971302, −24.963380929599044063195818592980, −23.8957002429788890790730956947, −22.38526032224391234553804965253, −20.64653162258021723690996015926, −18.88172009903748921865918078907, −18.10099096196119455815502387985, −16.15266427433562288425121751476, −14.28943082233952163374769888088, −12.7635194118700229837484632365, −11.459596005614086402668986184279, −9.03603034832864075846403276009, −7.35262170860794421983846811052, −5.554135368747861766165490725370, −2.43970471075016573834496104089, 3.53109568390976713881226562799, 5.37092217269348812234929796648, 7.86202871799724463684935052995, 9.885690281516151356044990603413, 11.001014977110963206286226455439, 13.27477102305485924531921767602, 14.96782345013431223635605146029, 16.30388879911594151964725727174, 17.69750908991989790359894140769, 19.882153214576594928835531512624, 20.93797208528347140576261539997, 22.434540191315472933521500272742, 23.7243438948056873765175022385, 25.77843933969612115054892329945, 26.78068812058721427987306146909, 28.047119757788737551702032536619, 29.504789230302340497825221713429, 31.1632075039371300308856175534, 32.570556274132070632133243141970, 33.45096533960072786896162409934, 34.849288868370123408737536992514, 36.63925162106026851035280910428, 37.64121840189302633480402058127, 39.336264703355623494329443153673, 39.68269158589923208615540280166

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