↑

Properties

Label 2-18e2-4.3-c2-0-1
Degree $2$
Conductor $324$
Sign $-0.500 + 0.866i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s − 4·5-s + 3.46i·7-s − 7.99·8-s + (−4 − 6.92i)10-s − 12.1i·11-s − 22·13-s + (−5.99 + 3.46i)14-s + (−8 − 13.8i)16-s + 11·17-s + 15.5i·19-s + (7.99 − 13.8i)20-s + (21 − 12.1i)22-s − 24.2i·23-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.800·5-s + 0.494i·7-s − 0.999·8-s + (−0.400 − 0.692i)10-s − 1.10i·11-s − 1.69·13-s + (−0.428 + 0.247i)14-s + (−0.5 − 0.866i)16-s + 0.647·17-s + 0.820i·19-s + (0.399 − 0.692i)20-s + (0.954 − 0.551i)22-s − 1.05i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.500 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.117869 - 0.204156i\)
\(L(\frac12)\) \(\approx\) \(0.117869 - 0.204156i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - 3.46iT - 49T^{2} \)
11 \( 1 + 12.1iT - 121T^{2} \)
13 \( 1 + 22T + 169T^{2} \)
17 \( 1 - 11T + 289T^{2} \)
19 \( 1 - 15.5iT - 361T^{2} \)
23 \( 1 + 24.2iT - 529T^{2} \)
29 \( 1 + 34T + 841T^{2} \)
31 \( 1 - 6.92iT - 961T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + 13T + 1.68e3T^{2} \)
43 \( 1 - 50.2iT - 1.84e3T^{2} \)
47 \( 1 - 3.46iT - 2.20e3T^{2} \)
53 \( 1 + 52T + 2.80e3T^{2} \)
59 \( 1 - 53.6iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 - 116. iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + 27.7iT - 6.24e3T^{2} \)
83 \( 1 - 34.6iT - 6.88e3T^{2} \)
89 \( 1 - 2T + 7.92e3T^{2} \)
97 \( 1 + 43T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.16687437668462091104790842322, −11.44408582024080247574252321488, −10.02136107865034756329437121277, −8.897995856692441783734792942107, −8.004856452859394810806702823993, −7.33531777423026067996840947870, −6.07554952496415258037269657421, −5.17682551419295047492980108134, −4.00017328884712627468985694417, −2.85065314871626626255400877561, 0.089793227576284730007959020522, 2.01908646439889452572414761392, 3.46032327380077039916406823055, 4.49697842583467922159930384649, 5.34816475989614953072269974695, 7.04981515771174708110289281762, 7.72604143941003819461615891172, 9.340798525142447161276401112844, 9.907240436237270001320970052939, 10.91998139205902570350720362843

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