L-function 1-45-45.32-r0-0-0

• Label: 1-45-45.32-r0-0-0
• Degree: $1$
• Conductor: $45$
• Sign: $0.746 + 0.665i$
• Analytic cond.: $0.208979$
• Root an. cond.: $0.208979$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(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 + ⋯

Functional equation

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

Invariants

Degree:	$1$
Conductor:	$45$    =    $3^{2} \cdot 5$
Sign:	$0.746 + 0.665i$
Analytic conductor:	$0.208979$
Root analytic conductor:	$0.208979$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{45} (32, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 45,\ (0:\ ),\ 0.746 + 0.665i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.121634195 + 0.4273851470i$
$L(1)$	$\approx$	$1.325722943 + 0.3763795750i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	5	$1$
good	2	$1 + (0.866 + 0.5i)T$
	7	$1 + (-0.866 - 0.5i)T$
	11	$1 + (0.5 - 0.866i)T$
	13	$1 + (-0.866 + 0.5i)T$
	17	$1 - iT$
	19	$1 - T$
	23	$1 + (0.866 - 0.5i)T$
	29	$1 + (-0.5 + 0.866i)T$
	31	$1 + (-0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

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

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