Properties

Degree 1
Conductor 73
Sign $-0.928 + 0.371i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.984 − 0.173i)5-s + (−0.766 + 0.642i)6-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s i·10-s + (0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (0.642 + 0.766i)13-s + (−0.642 − 0.766i)14-s + (−0.342 − 0.939i)15-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)17-s + ⋯
L(s,χ)  = 1  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.984 − 0.173i)5-s + (−0.766 + 0.642i)6-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s i·10-s + (0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (0.642 + 0.766i)13-s + (−0.642 − 0.766i)14-s + (−0.342 − 0.939i)15-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.928 + 0.371i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.928 + 0.371i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $-0.928 + 0.371i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (48, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ -0.928 + 0.371i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1610255237 + 0.8350704434i$
$L(\frac12,\chi)$  $\approx$  $0.1610255237 + 0.8350704434i$
$L(\chi,1)$  $\approx$  0.5961649543 + 0.7498615877i
$L(1,\chi)$  $\approx$  0.5961649543 + 0.7498615877i

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

−30.816293475921249287052077985008, −30.053327434363477076146785974580, −29.325929163491932273419394286054, −27.91358437480305033927113489011, −26.93914149233832717876575387430, −25.7587878242011741640503498050, −24.35993274301470286674564027467, −22.93747414549045675507715385716, −22.79277791027564704995342294056, −20.726219373064267941067135745780, −19.893603830916183739030603524997, −19.12237476554411299571046648524, −18.29580642605189862323075074136, −16.63260306798926527676816255303, −14.851970406681215404723748798277, −13.83329590070698418655270259842, −12.6250134299873760047208616088, −11.876030537216652028591094854758, −10.44085414396535789094706737001, −8.95632221424495979420098911189, −7.74150926299161185182749546096, −6.20318820424709341598503538729, −3.87926348497794108570691518829, −3.07703637070973490384758155346, −0.991251142245202575360958513907, 3.46121254528494352740723149142, 4.2875987380435045054808662486, 5.92511871408424527986267994648, 7.452292253174317845121889197649, 8.81961643687586543036021658784, 9.502767806878153953164544415554, 11.49345789824953476548866903073, 12.94674752346500643389884477658, 14.356162206126585731954229834026, 15.337299168811905885129552971866, 16.10025012259478842250902321873, 16.981390845855785789764844568377, 18.91062084289174967078563543687, 19.68988623221941470141048581714, 21.33038248058410902319218031909, 22.332433514100235702750103948201, 23.25211397226014495746179871900, 24.52115823083133363635512499322, 25.76607668819732482496391677194, 26.2866026779685527054227819046, 27.72485678520817087340671318645, 28.05787912460182457452769417661, 30.37977035044654978620640677673, 31.41227814820123555081345284068, 32.1111505299062050968237736107

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