Properties

Label 1-18e2-324.275-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: $\chi_{324} (275, \cdot )$
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

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

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