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Properties

Label 2-24e2-144.61-c1-0-2
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

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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: $\chi_{576} (529, \cdot )$
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 \( 1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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