Properties

Label 2-648-648.139-c0-0-0
Degree $2$
Conductor $648$
Sign $-0.790 + 0.612i$
Analytic cond. $0.323394$
Root an. cond. $0.568677$
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)2-s + (−0.0581 − 0.998i)3-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)6-s + (−0.939 − 0.342i)8-s + (−0.993 + 0.116i)9-s + (−0.512 − 0.257i)11-s + (−0.939 + 0.342i)12-s + (−0.835 + 0.549i)16-s + (0.606 + 0.509i)17-s + (−0.499 + 0.866i)18-s + (1.36 − 1.14i)19-s + (−0.512 + 0.257i)22-s + (−0.286 + 0.957i)24-s + (0.396 + 0.918i)25-s + ⋯
L(s)  = 1  + (0.597 − 0.802i)2-s + (−0.0581 − 0.998i)3-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)6-s + (−0.939 − 0.342i)8-s + (−0.993 + 0.116i)9-s + (−0.512 − 0.257i)11-s + (−0.939 + 0.342i)12-s + (−0.835 + 0.549i)16-s + (0.606 + 0.509i)17-s + (−0.499 + 0.866i)18-s + (1.36 − 1.14i)19-s + (−0.512 + 0.257i)22-s + (−0.286 + 0.957i)24-s + (0.396 + 0.918i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ -0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.113447824\)
\(L(\frac12)\) \(\approx\) \(1.113447824\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.597 + 0.802i)T \)
3 \( 1 + (0.0581 + 0.998i)T \)
good5 \( 1 + (-0.396 - 0.918i)T^{2} \)
7 \( 1 + (0.0581 - 0.998i)T^{2} \)
11 \( 1 + (0.512 + 0.257i)T + (0.597 + 0.802i)T^{2} \)
13 \( 1 + (0.686 + 0.727i)T^{2} \)
17 \( 1 + (-0.606 - 0.509i)T + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (-1.36 + 1.14i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.0581 + 0.998i)T^{2} \)
29 \( 1 + (-0.973 - 0.230i)T^{2} \)
31 \( 1 + (-0.893 - 0.448i)T^{2} \)
37 \( 1 + (0.939 + 0.342i)T^{2} \)
41 \( 1 + (0.0694 + 0.0932i)T + (-0.286 + 0.957i)T^{2} \)
43 \( 1 + (0.0694 + 1.19i)T + (-0.993 + 0.116i)T^{2} \)
47 \( 1 + (-0.893 + 0.448i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (1.49 - 0.749i)T + (0.597 - 0.802i)T^{2} \)
61 \( 1 + (0.835 + 0.549i)T^{2} \)
67 \( 1 + (-1.36 + 0.159i)T + (0.973 - 0.230i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.86 - 0.679i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.286 + 0.957i)T^{2} \)
83 \( 1 + (-0.207 + 0.278i)T + (-0.286 - 0.957i)T^{2} \)
89 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (1.62 - 1.06i)T + (0.396 - 0.918i)T^{2} \)
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

−10.79746293678126620137931071760, −9.645924482454335245697978543313, −8.800233988342310190503709205756, −7.69107837932426006034314667394, −6.78774046511568001612945096841, −5.69230048177721342257231823228, −5.05609490657813513837488820604, −3.47395866007221958681305328082, −2.57520447454442846389737597677, −1.18272167151859846625180239184, 2.86034020251550513474836739733, 3.75689535633358187668560010309, 4.87406046446822597067747504786, 5.47390610116312115380121974118, 6.46758723870284068637349936226, 7.69265198737071884622349342116, 8.301418561659949652843061119278, 9.465270638420113971307864587265, 10.02088608470243839351918782122, 11.18092270381328031280733654978

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