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Properties

Label 1-18e2-324.11-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  + (−0.973 − 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

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

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: $\chi_{324} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ 0.999 + 0.0193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005182574 + 0.009746830739i\)
\(L(\frac12)\) \(\approx\) \(1.005182574 + 0.009746830739i\)
\(L(1)\) \(\approx\) \(0.9125371371 + 0.002206512224i\)
\(L(1)\) \(\approx\) \(0.9125371371 + 0.002206512224i\)

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

−25.21185699760002073780104378655, −23.92715529413238594882861400271, −23.18812042485388868224737714666, −22.85497115732654850936067683836, −21.51261694638207476007931367230, −20.41841328743376218705389663858, −19.873412913563219770631436925754, −18.79578091638433377848734386891, −18.143780240046855260913833077119, −16.72311002752624189637015352751, −16.13175074735818207121344434286, −15.21013433164945481999068361121, −14.2160409647407043396488272207, −13.13138103383540902581860369994, −12.30709566942942161995645507885, −11.17101734045921546241288471596, −10.4041336290641915589660450268, −9.32723219965925425527014591863, −7.99287907397304247260976812670, −7.30196911853906386865361190507, −6.29973864277110199903937959078, −4.78789975635686672655593888270, −3.81383913594942986503413929944, −2.8570222036112290404868096108, −0.96446388521425799459019478725, 0.948739627459438806100274307052, 2.95108746644312063659509765430, 3.55754562032854259076253112583, 5.15488882452516813642923991179, 5.92799640287138050403313872444, 7.3553312805279909571608629078, 8.242443940519707028696777918808, 9.09399602990853329732985211275, 10.323971799458279585298386768464, 11.413463060607389859342221364970, 12.19124863931265827795321405713, 13.09754677392290713417694988548, 14.1355421166074370129051749985, 15.61696044880926365253620024952, 15.738483233945319744848906371947, 16.74348527095360657301573892034, 18.23118686186212640074173647008, 18.80352949306006249202483636899, 19.62738524142504913391977524810, 20.67841983194909200173705067340, 21.458272009938699952414729979417, 22.640081422783803024435586471975, 23.25545434475546726712372577828, 24.13218969521705289181043897628, 25.1633408374727157430735329941

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