Properties

Label 1-18e2-324.119-r0-0-0
Degree $1$
Conductor $324$
Sign $0.999 + 0.0193i$
Analytic cond. $1.50464$
Root an. cond. $1.50464$
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  + (0.686 − 0.727i)5-s + (0.993 + 0.116i)7-s + (−0.286 + 0.957i)11-s + (−0.0581 + 0.998i)13-s + (0.939 − 0.342i)17-s + (0.939 + 0.342i)19-s + (−0.993 + 0.116i)23-s + (−0.0581 − 0.998i)25-s + (−0.893 − 0.448i)29-s + (−0.597 − 0.802i)31-s + (0.766 − 0.642i)35-s + (0.766 + 0.642i)37-s + (0.835 + 0.549i)41-s + (−0.973 + 0.230i)43-s + (0.597 − 0.802i)47-s + ⋯
L(s)  = 1  + (0.686 − 0.727i)5-s + (0.993 + 0.116i)7-s + (−0.286 + 0.957i)11-s + (−0.0581 + 0.998i)13-s + (0.939 − 0.342i)17-s + (0.939 + 0.342i)19-s + (−0.993 + 0.116i)23-s + (−0.0581 − 0.998i)25-s + (−0.893 − 0.448i)29-s + (−0.597 − 0.802i)31-s + (0.766 − 0.642i)35-s + (0.766 + 0.642i)37-s + (0.835 + 0.549i)41-s + (−0.973 + 0.230i)43-s + (0.597 − 0.802i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.999 + 0.0193i$
Analytic conductor: \(1.50464\)
Root analytic conductor: \(1.50464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ 0.999 + 0.0193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.584643724 + 0.01536562070i\)
\(L(\frac12)\) \(\approx\) \(1.584643724 + 0.01536562070i\)
\(L(1)\) \(\approx\) \(1.298619078 + 0.02870540842i\)
\(L(1)\) \(\approx\) \(1.298619078 + 0.02870540842i\)

Euler product

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

−25.07063021216403584269590137463, −24.26552201702305944013503230024, −23.38133231601688770397715934055, −22.253451051999472152804379664083, −21.61516623949297683838541444222, −20.76991079535852483123735795923, −19.813201737351551387377425420614, −18.50024691367706226372540450935, −18.08004052236392394259539218078, −17.14517555366707383544775341215, −16.080437390072314296460631226382, −14.90247290259134507384454952059, −14.20635614463951039749754020935, −13.443089740682811863615052200374, −12.208092299812950278162699877635, −10.998613430260996288121094700145, −10.50904754153473315979119877738, −9.33238039674259874775732197852, −8.07282546362385706454761879116, −7.34718279025941008196226905610, −5.83595600416796153224316613111, −5.34522789386644298633810726300, −3.66380804545492607044899776019, −2.63359579065309665679371310974, −1.28146615505699030615773510310, 1.42578100410181269638867960062, 2.24943412176627100018912240941, 4.08979038333816930770786681624, 5.037159982820941948716577361777, 5.860827259244695871130308492211, 7.35143276115131161549563845092, 8.1811373002495611318623721455, 9.439891294058329324591960927122, 9.974858720463565560359235759137, 11.495432925678721486098201368977, 12.13463868994984352739277606540, 13.282752181978327044949684016751, 14.17294210343459481626267668317, 14.956461782241177294760678197, 16.2797584622301731151892081235, 16.94345536523056374369929997995, 17.99197071686578726179073152759, 18.52589428018255920049614562316, 20.05014051973512828333297776556, 20.69953020870733025717040304512, 21.35817228832665164414593856733, 22.32494354226508727841378077226, 23.55208683525819447888939379452, 24.21227706940565031890860714931, 25.04670926711771596877237525349

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