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Properties

Label 1-45-45.38-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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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: $\chi_{45} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 45,\ (0:\ ),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.121634195 - 0.4273851470i\)
\(L(\frac12)\) \(\approx\) \(1.121634195 - 0.4273851470i\)
\(L(1)\) \(\approx\) \(1.325722943 - 0.3763795750i\)
\(L(1)\) \(\approx\) \(1.325722943 - 0.3763795750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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