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Properties

Label 1-18e2-324.103-r1-0-0
Degree $1$
Conductor $324$
Sign $0.0193 - 0.999i$
Analytic cond. $34.8186$
Root an. cond. $34.8186$
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  + (0.973 − 0.230i)5-s + (−0.597 − 0.802i)7-s + (0.686 − 0.727i)11-s + (0.893 + 0.448i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (−0.597 + 0.802i)23-s + (0.893 − 0.448i)25-s + (−0.835 − 0.549i)29-s + (−0.396 − 0.918i)31-s + (−0.766 − 0.642i)35-s + (0.766 − 0.642i)37-s + (−0.0581 − 0.998i)41-s + (0.286 + 0.957i)43-s + (−0.396 + 0.918i)47-s + ⋯
L(s)  = 1  + (0.973 − 0.230i)5-s + (−0.597 − 0.802i)7-s + (0.686 − 0.727i)11-s + (0.893 + 0.448i)13-s + (−0.939 − 0.342i)17-s + (0.939 − 0.342i)19-s + (−0.597 + 0.802i)23-s + (0.893 − 0.448i)25-s + (−0.835 − 0.549i)29-s + (−0.396 − 0.918i)31-s + (−0.766 − 0.642i)35-s + (0.766 − 0.642i)37-s + (−0.0581 − 0.998i)41-s + (0.286 + 0.957i)43-s + (−0.396 + 0.918i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.0193 - 0.999i$
Analytic conductor: \(34.8186\)
Root analytic conductor: \(34.8186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (1:\ ),\ 0.0193 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.488912309 - 1.460314899i\)
\(L(\frac12)\) \(\approx\) \(1.488912309 - 1.460314899i\)
\(L(1)\) \(\approx\) \(1.202707847 - 0.3490021506i\)
\(L(1)\) \(\approx\) \(1.202707847 - 0.3490021506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.973 - 0.230i)T \)
7 \( 1 + (-0.597 - 0.802i)T \)
11 \( 1 + (0.686 - 0.727i)T \)
13 \( 1 + (0.893 + 0.448i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.597 + 0.802i)T \)
29 \( 1 + (-0.835 - 0.549i)T \)
31 \( 1 + (-0.396 - 0.918i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.0581 - 0.998i)T \)
43 \( 1 + (0.286 + 0.957i)T \)
47 \( 1 + (-0.396 + 0.918i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.0581 - 0.998i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.973 + 0.230i)T \)
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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

−25.16705563898010576470554295033, −24.50018368710282541741502247957, −23.123101378242419672337503133912, −22.20537186812576328438259905860, −21.86244811349647381298079857309, −20.5470286136651627537250630301, −19.91314306736681334077061755061, −18.46446267847686888604590732889, −18.150849151409719048488626447501, −17.04945260637693914156085222533, −16.036512250319426151181585063618, −15.09028477403827498578262827667, −14.17720617264286196406735636679, −13.14408707715683678529065010248, −12.419872034494203050917783465126, −11.22069999929386895071954312700, −10.11019595473196351291155281514, −9.33120288381982549023008704328, −8.461075213402375167815904626936, −6.87267131117470315454896036398, −6.15911522642481921832788839706, −5.209015690104585493028846972435, −3.71079005533466657158991533510, −2.513547855170491131186051525877, −1.428638265160529711488619139759, 0.6190199955139622677954725679, 1.82047312521555045757009015859, 3.3112929916594135100406140234, 4.336241824887828179703756392277, 5.80912720964646003477502463955, 6.45118030809804991352363091760, 7.628096323526739146630321078315, 9.13724084243130323566718157582, 9.505854524991439916550935244071, 10.815881683024932061356645541492, 11.59795190903501198177216590691, 13.1561024170946137298055422845, 13.548649544675597224730865349589, 14.355348241750872511641362085287, 15.87958702188152374026351539226, 16.51507887951319678978837920162, 17.42862912315677793449616059077, 18.25987045851743145893546669218, 19.38549379420709056366326324306, 20.2402010051291019179926781112, 21.06656240007716092876209324085, 22.09177375066047654497324307587, 22.67663340482275679908802338169, 23.991124705726708387803978230723, 24.53665335311585843542018953149

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