Properties

Degree 1
Conductor $ 3^{2} \cdot 5 $
Sign $0.746 - 0.665i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s i·8-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + i·17-s − 19-s + (0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s − 26-s + i·28-s + (−0.5 − 0.866i)29-s + ⋯
L(s,χ)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s i·8-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + i·17-s − 19-s + (0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s − 26-s + i·28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.746 - 0.665i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.746 - 0.665i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(45\)    =    \(3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.746 - 0.665i$
motivic weight  =  \(0\)
character  :  $\chi_{45} (38, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 45,\ (0:\ ),\ 0.746 - 0.665i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.121634195 - 0.4273851470i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.121634195 - 0.4273851470i\)
\(L(\chi,1)\)  \(\approx\)  \(1.325722943 - 0.3763795750i\)
\(L(1,\chi)\)  \(\approx\)  \(1.325722943 - 0.3763795750i\)

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

−34.35120427433898782995849327214, −33.164274991087520294780567206496, −32.17900424643717099887851175528, −31.34730036269974033254106837475, −29.75946837397514624139563632631, −29.241168675169633748227985056820, −27.16101443131673720727628981936, −26.10108139965325163671333111158, −24.922872477428019857450742488324, −23.8332839457911888648774790342, −22.65417322400857212273126914082, −21.74422327324655778294265741098, −20.34043452267309477114702265649, −19.02740114767636611270912008177, −17.00476420403583707246790406379, −16.32354548153577964310638340464, −14.79628171318518435844216429572, −13.65955531716316586034419374885, −12.50645745592483848194158297560, −11.07297458220718648110346385663, −9.14276701251768334339005258195, −7.32091131843316737551159368428, −6.19824974543228993050043173011, −4.46770315626047345629366907824, −2.95556757703445041003603393133, 2.27801123733464486006762319605, 3.928016253803500703636670860979, 5.58872653978057000220107781450, 6.981415230962860811795010775574, 9.33138809756658955992612103916, 10.57842653171016829805642816678, 12.2586821272709000813693138281, 12.933750872201971845870702291323, 14.65619229002377086451959002444, 15.51854160130533112352736896324, 17.22470879883436590376710263106, 19.0942840596746855877614191987, 19.83993267788212567375159670710, 21.3221358381869911538380189064, 22.35187144111126316973226425094, 23.26671817717762986731123883769, 24.72183463160220183854718004095, 25.68879787270294708197101095750, 27.59103933300739950094295491547, 28.61826334367469613993085645601, 29.682853874499062845660019833697, 30.765176794437272535741646955643, 31.94992297959395786946267085022, 32.706828236563411129643526471574, 34.00424710182707550064941062131

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