Properties

Label 2-24e2-144.85-c1-0-18
Degree $2$
Conductor $576$
Sign $-0.216 + 0.976i$
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  + (1.5 − 0.866i)3-s + (−3.73 + i)5-s + (0.633 − 0.366i)7-s + (1.5 − 2.59i)9-s + (0.767 − 2.86i)11-s + (−1.63 − 6.09i)13-s + (−4.73 + 4.73i)15-s − 2.26·17-s + (0.633 − 0.633i)19-s + (0.633 − 1.09i)21-s + (1.09 + 0.633i)23-s + (8.59 − 4.96i)25-s − 5.19i·27-s + (−2.36 − 0.633i)29-s + (3.73 − 6.46i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−1.66 + 0.447i)5-s + (0.239 − 0.138i)7-s + (0.5 − 0.866i)9-s + (0.231 − 0.864i)11-s + (−0.453 − 1.69i)13-s + (−1.22 + 1.22i)15-s − 0.550·17-s + (0.145 − 0.145i)19-s + (0.138 − 0.239i)21-s + (0.228 + 0.132i)23-s + (1.71 − 0.992i)25-s − 0.999i·27-s + (−0.439 − 0.117i)29-s + (0.670 − 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.216 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776903 - 0.968001i\)
\(L(\frac12)\) \(\approx\) \(0.776903 - 0.968001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
good5 \( 1 + (3.73 - i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.633 + 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.767 + 2.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.63 + 6.09i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + (-0.633 + 0.633i)T - 19iT^{2} \)
23 \( 1 + (-1.09 - 0.633i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.36 + 0.633i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.73 + 6.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 1.26i)T + 37iT^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.330 - 1.23i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.83 - 8.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.535 + 0.535i)T + 53iT^{2} \)
59 \( 1 + (4.96 - 1.33i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3 + 0.803i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.40 + 5.23i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.36 - 0.366i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (4.13 + 7.16i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.67587825562507321789220690126, −9.438693655527150830481389603152, −8.329730102816591690684287999116, −7.88151414960604776218478956243, −7.26120610036209112453457874834, −6.10517468183462305665122457970, −4.52894018041374925835526999335, −3.49327055423267326069165901655, −2.79285429412326055230635552274, −0.64718781797248432170085297969, 1.93184635932870062166391627912, 3.46159142506378581515616693618, 4.41212321747056481783418730999, 4.80262701270519440538109749946, 6.91859964322026948445517198573, 7.45522003395365809876340972527, 8.523121281565423988337395873083, 8.971112099007622635987540572515, 9.949732830602135003373638859706, 11.07950360506339354164715831463

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