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Properties

Label 1-18e2-324.187-r1-0-0
Degree $1$
Conductor $324$
Sign $-0.627 + 0.778i$
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  + (−0.835 − 0.549i)5-s + (0.686 + 0.727i)7-s + (−0.893 − 0.448i)11-s + (0.396 − 0.918i)13-s + (0.766 + 0.642i)17-s + (−0.766 + 0.642i)19-s + (0.686 − 0.727i)23-s + (0.396 + 0.918i)25-s + (−0.993 − 0.116i)29-s + (−0.973 − 0.230i)31-s + (−0.173 − 0.984i)35-s + (0.173 − 0.984i)37-s + (0.597 + 0.802i)41-s + (0.0581 + 0.998i)43-s + (−0.973 + 0.230i)47-s + ⋯
L(s)  = 1  + (−0.835 − 0.549i)5-s + (0.686 + 0.727i)7-s + (−0.893 − 0.448i)11-s + (0.396 − 0.918i)13-s + (0.766 + 0.642i)17-s + (−0.766 + 0.642i)19-s + (0.686 − 0.727i)23-s + (0.396 + 0.918i)25-s + (−0.993 − 0.116i)29-s + (−0.973 − 0.230i)31-s + (−0.173 − 0.984i)35-s + (0.173 − 0.984i)37-s + (0.597 + 0.802i)41-s + (0.0581 + 0.998i)43-s + (−0.973 + 0.230i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.627 + 0.778i$
Analytic conductor: \(34.8186\)
Root analytic conductor: \(34.8186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (1:\ ),\ -0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2367105507 + 0.4950381224i\)
\(L(\frac12)\) \(\approx\) \(0.2367105507 + 0.4950381224i\)
\(L(1)\) \(\approx\) \(0.8153448387 + 0.02681376204i\)
\(L(1)\) \(\approx\) \(0.8153448387 + 0.02681376204i\)

Euler product

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

−24.28184026477672237073635910684, −23.45523723718001283809876143228, −23.18857689838650901240648241216, −21.85740320309631047472131291191, −20.8929809187386091181854600450, −20.1933448158573010267302392559, −19.04990131703600133142101742616, −18.43721738289357990127972961986, −17.362072555450632138201771184342, −16.38595056272042796657874936349, −15.42048522188181435683618008121, −14.60312235037553250242240942862, −13.70046784429229162607261022365, −12.60675253239859261454893304913, −11.33789069164166462441642652885, −10.96575383830513420945791758210, −9.7600580330899526927724726652, −8.45494516044810107823946229046, −7.45237851379660291313542947004, −6.90740887164601073820785485295, −5.24235374041068314805236433287, −4.26909426246725652233336781189, −3.22557888878510372237253281492, −1.78450245031963679935765009289, −0.16731878595604779112232689421, 1.29074264164803493572291144331, 2.802612579635217502699690087043, 3.998774324482776134294775417342, 5.19443803148038572923364002593, 5.94774390158343531721861606027, 7.77934656991562161155561155061, 8.12800895977203051178614774260, 9.14248261148063679045356753829, 10.654692540567895600095929446409, 11.27357403695338182212806441912, 12.59163067708482282632073888708, 12.902293340508357274340200461381, 14.58443661630542692026698572629, 15.16721283814660299452154734198, 16.14361679632088198616451125555, 16.95217140638646965575905346776, 18.23546398783708773261957384746, 18.81203961805803725748889749638, 19.86728259555737363530788430444, 20.92086231339296803882980806514, 21.320970026541376324691969249886, 22.745185110839667552430033762818, 23.44749976486018153985558858866, 24.30893996222053764288610414904, 25.02457926256744035770674451541

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