↑

Properties

Label 2-24e2-9.7-c1-0-8
Degree $2$
Conductor $576$
Sign $0.766 + 0.642i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  − 1.73·3-s + (−0.5 − 0.866i)5-s + (−0.866 + 1.5i)7-s + 2.99·9-s + (−0.866 + 1.5i)11-s + (−1.5 − 2.59i)13-s + (0.866 + 1.49i)15-s + 4·17-s + 6.92·19-s + (1.49 − 2.59i)21-s + (−4.33 − 7.5i)23-s + (2 − 3.46i)25-s − 5.19·27-s + (0.5 − 0.866i)29-s + (2.59 + 4.5i)31-s + ⋯
L(s)  = 1  − 1.00·3-s + (−0.223 − 0.387i)5-s + (−0.327 + 0.566i)7-s + 0.999·9-s + (−0.261 + 0.452i)11-s + (−0.416 − 0.720i)13-s + (0.223 + 0.387i)15-s + 0.970·17-s + 1.58·19-s + (0.327 − 0.566i)21-s + (−0.902 − 1.56i)23-s + (0.400 − 0.692i)25-s − 1.00·27-s + (0.0928 − 0.160i)29-s + (0.466 + 0.808i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871048 - 0.317035i\)
\(L(\frac12)\) \(\approx\) \(0.871048 - 0.317035i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.866 - 1.5i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + (4.33 + 7.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.59 - 4.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.06 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 7.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (2.59 - 4.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.33 + 7.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.29230200979393891471279956020, −10.18296665693733248169173399039, −8.920837932191506036001923355549, −7.84165846407462621901386814862, −7.01020004229341744030646703982, −5.82632683410062399018216428840, −5.23770239913816992599115931990, −4.17145312869227805869359197685, −2.64210390742601248164174795270, −0.74824771779610708114160593768, 1.14955215450611724539340734121, 3.15688185725930007471560628015, 4.23725515844151126094460315478, 5.42345236604146557386579776974, 6.17815024321373516524761835285, 7.37590504463993109679122082694, 7.68473039817180142527553158022, 9.547874672820915986684774364673, 9.857598005828976264418580724321, 11.01221121458362073905798964412

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