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Properties

Label 2-24e2-144.85-c1-0-4
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  + (−0.866 − 1.5i)3-s + (1 − 0.267i)5-s + (−2.36 + 1.36i)7-s + (−1.5 + 2.59i)9-s + (−1.13 + 4.23i)11-s + (0.901 + 3.36i)13-s + (−1.26 − 1.26i)15-s − 5.73·17-s + (2.36 − 2.36i)19-s + (4.09 + 2.36i)21-s + (4.09 + 2.36i)23-s + (−3.40 + 1.96i)25-s + 5.19·27-s + (2.36 + 0.633i)29-s + (0.267 − 0.464i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (0.447 − 0.119i)5-s + (−0.894 + 0.516i)7-s + (−0.5 + 0.866i)9-s + (−0.341 + 1.27i)11-s + (0.250 + 0.933i)13-s + (−0.327 − 0.327i)15-s − 1.39·17-s + (0.542 − 0.542i)19-s + (0.894 + 0.516i)21-s + (0.854 + 0.493i)23-s + (−0.680 + 0.392i)25-s + 1.00·27-s + (0.439 + 0.117i)29-s + (0.0481 − 0.0833i)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

\(L(1)\) \(\approx\) \(0.596761 + 0.478951i\)
\(L(\frac12)\) \(\approx\) \(0.596761 + 0.478951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
good5 \( 1 + (-1 + 0.267i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.13 - 4.23i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.901 - 3.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (-2.36 + 2.36i)T - 19iT^{2} \)
23 \( 1 + (-4.09 - 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.36 - 0.633i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.267 + 0.464i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 4.73i)T + 37iT^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.23 - 8.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.83 + 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.46 + 7.46i)T + 53iT^{2} \)
59 \( 1 + (7.33 - 1.96i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-11.1 - 3i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.76 - 6.59i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
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 \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.18406458291459731609650440071, −9.848035852700169833864911629542, −9.316300470933967208490599295751, −8.228027960622964504264802193992, −6.90700348871671951527275815094, −6.67603951010784548686633086492, −5.49537223598196121522978518526, −4.55651518398104424197820009672, −2.75544305700835246477791237629, −1.71877354122157718157105671499, 0.44716466961339725046195303942, 2.89993379860249389600465731415, 3.74179055227124784312286030819, 4.99844520234877892242828333485, 6.02083944492527300538624081356, 6.53442058697239782247019023900, 8.001089553094951181872760568797, 9.009223649795618703342731831872, 9.768577686848571419976274451158, 10.68433457626288636818750729249

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