↑

Properties

Label 2-6e3-27.16-c1-0-5
Degree $2$
Conductor $216$
Sign $0.847 - 0.531i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.997 + 1.41i)3-s + (1.95 + 0.711i)5-s + (0.739 − 4.19i)7-s + (−1.01 + 2.82i)9-s + (2.33 − 0.848i)11-s + (−4.38 + 3.67i)13-s + (0.941 + 3.47i)15-s + (0.340 + 0.589i)17-s + (−2.58 + 4.47i)19-s + (6.67 − 3.13i)21-s + (−1.23 − 6.99i)23-s + (−0.514 − 0.431i)25-s + (−5.00 + 1.38i)27-s + (4.22 + 3.54i)29-s + (−0.787 − 4.46i)31-s + ⋯
L(s)  = 1  + (0.575 + 0.817i)3-s + (0.874 + 0.318i)5-s + (0.279 − 1.58i)7-s + (−0.337 + 0.941i)9-s + (0.702 − 0.255i)11-s + (−1.21 + 1.01i)13-s + (0.243 + 0.898i)15-s + (0.0825 + 0.142i)17-s + (−0.592 + 1.02i)19-s + (1.45 − 0.684i)21-s + (−0.257 − 1.45i)23-s + (−0.102 − 0.0862i)25-s + (−0.963 + 0.266i)27-s + (0.785 + 0.658i)29-s + (−0.141 − 0.802i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.847 - 0.531i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.847 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53320 + 0.441162i\)
\(L(\frac12)\) \(\approx\) \(1.53320 + 0.441162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.997 - 1.41i)T \)
good5 \( 1 + (-1.95 - 0.711i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.739 + 4.19i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.33 + 0.848i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (4.38 - 3.67i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.340 - 0.589i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.58 - 4.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 + 6.99i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-4.22 - 3.54i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.787 + 4.46i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.69 + 6.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.256 - 0.214i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.61 - 0.586i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.889 - 5.04i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 9.45T + 53T^{2} \)
59 \( 1 + (9.52 + 3.46i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.36 + 7.73i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.13 + 2.63i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.77 - 4.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.43 - 7.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.53 - 3.80i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.90 + 5.79i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-0.983 + 1.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.443 + 0.161i)T + (74.3 - 62.3i)T^{2} \)
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

−12.47311151429868124063237582021, −11.08778380854852978999126777841, −10.25526685184010058320685979053, −9.754407955802798569997127670015, −8.581378873951766025171805094665, −7.38382852811705043023454043883, −6.30990425430138034940266757592, −4.66834676169147764106789971228, −3.85241057070572924127471513461, −2.11466539330180175482149958266, 1.85371974708806059088319540773, 2.88425085081613114307131235939, 5.10619883875876765401743949396, 5.99255648042061835757960203154, 7.19823566313855430560030228035, 8.420433263755412988754427852740, 9.158050378993931602284942751850, 9.946211897464049805302917919302, 11.75327042280878132024210840529, 12.22041342549297321549501827366

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