↑

Properties

Label 1-18e2-324.223-r1-0-0
Degree $1$
Conductor $324$
Sign $0.856 - 0.516i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.597 + 0.802i)5-s + (0.835 − 0.549i)7-s + (0.993 + 0.116i)11-s + (−0.286 − 0.957i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)19-s + (0.835 + 0.549i)23-s + (−0.286 + 0.957i)25-s + (−0.686 − 0.727i)29-s + (0.0581 − 0.998i)31-s + (0.939 + 0.342i)35-s + (−0.939 + 0.342i)37-s + (0.973 + 0.230i)41-s + (−0.396 − 0.918i)43-s + (0.0581 + 0.998i)47-s + ⋯
L(s)  = 1  + (0.597 + 0.802i)5-s + (0.835 − 0.549i)7-s + (0.993 + 0.116i)11-s + (−0.286 − 0.957i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)19-s + (0.835 + 0.549i)23-s + (−0.286 + 0.957i)25-s + (−0.686 − 0.727i)29-s + (0.0581 − 0.998i)31-s + (0.939 + 0.342i)35-s + (−0.939 + 0.342i)37-s + (0.973 + 0.230i)41-s + (−0.396 − 0.918i)43-s + (0.0581 + 0.998i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(34.8186\)
Root analytic conductor: \(34.8186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (1:\ ),\ 0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.458500308 - 0.6843703352i\)
\(L(\frac12)\) \(\approx\) \(2.458500308 - 0.6843703352i\)
\(L(1)\) \(\approx\) \(1.404037744 - 0.09939165699i\)
\(L(1)\) \(\approx\) \(1.404037744 - 0.09939165699i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−24.751459041509830735583113836131, −24.35067725751082075823182141352, −23.31343666017606667066592181472, −22.02728663775591339428006963475, −21.361353202695709702739566171486, −20.71622224436641812446895879931, −19.569927752953982052770827306601, −18.71319157700830233949994426178, −17.56634253173585255845272255710, −16.93641776721718936929998260528, −16.11588477637011635718674038, −14.54114990356812863989010910692, −14.37950604250172473462988686207, −12.89096187270413183136871684648, −12.17475442215925858124719005811, −11.22507962160270993211894188747, −9.98836024985025103305914082549, −8.89192724913516839345881279697, −8.4315565505283629034843550530, −6.90143673660559803800816416839, −5.81339277413162334846471376975, −4.88312688254857910485194251904, −3.82372374734698359077406129884, −2.01910912090980888109019797375, −1.30838412466697190990411593908, 0.82300163930312172796738865707, 2.18161548480758147590623921976, 3.34844925983489584712005664184, 4.66719324247333924405530469830, 5.738095755716486175628386278563, 6.97255825394665471609485789835, 7.62672759706143801706052918761, 9.07394126288947800159599044688, 9.95640936694428610321319575915, 11.01700958613991590230799723867, 11.63489364729822481391897380417, 13.11147446657679126949127298892, 13.93372937237403830125967792992, 14.72968358392947906647042762824, 15.503368776271292031339690458564, 17.179867672626109702063175179444, 17.37275938887727912135473908208, 18.42731334625520995865505860958, 19.4296693085496734116376117911, 20.43762488958259474506919331466, 21.20047426119191728644798281841, 22.340085608508687139133956625173, 22.75055547291619662949331672538, 24.0022422409151798426225339946, 24.87977498067057872222875338368

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