Properties

Label 2-24e2-9.4-c1-0-5
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  − 1.73i·3-s + (−0.5 + 0.866i)5-s + (1.5 + 2.59i)7-s − 2.99·9-s + (2.5 + 4.33i)11-s + (−2.5 + 4.33i)13-s + (1.49 + 0.866i)15-s − 2·17-s + 4·19-s + (4.5 − 2.59i)21-s + (0.5 − 0.866i)23-s + (2 + 3.46i)25-s + 5.19i·27-s + (−4.5 − 7.79i)29-s + (0.5 − 0.866i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.223 + 0.387i)5-s + (0.566 + 0.981i)7-s − 0.999·9-s + (0.753 + 1.30i)11-s + (−0.693 + 1.20i)13-s + (0.387 + 0.223i)15-s − 0.485·17-s + 0.917·19-s + (0.981 − 0.566i)21-s + (0.104 − 0.180i)23-s + (0.400 + 0.692i)25-s + 0.999i·27-s + (−0.835 − 1.44i)29-s + (0.0898 − 0.155i)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: $\chi_{576} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25382 + 0.456355i\)
\(L(\frac12)\) \(\approx\) \(1.25382 + 0.456355i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.35189272500280184237064208959, −9.655315308619979168593484329107, −9.130423303298720833791456955605, −7.987446161744853082248826737896, −7.18959979732730806452030239329, −6.53189690748097713866512049406, −5.36825962588137218970405058696, −4.25896136730717251019945064395, −2.56771223640190494402709759702, −1.73617124899031196051061078570, 0.796048099372776484310285925970, 3.06682822576520482311985729038, 3.96432099671341392538325693975, 4.95399701410627557008623306328, 5.76001652068867621954264549495, 7.15647178722108643884658545671, 8.166221704813439176829033591306, 8.862585617961254780740691537800, 9.819470853695833479821261566070, 10.72893673782673732821568918200

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