Properties

Label 2-24e2-576.205-c1-0-84
Degree $2$
Conductor $576$
Sign $-0.309 + 0.950i$
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  + (1.08 + 0.905i)2-s + (0.0543 − 1.73i)3-s + (0.360 + 1.96i)4-s + (−2.37 − 2.70i)5-s + (1.62 − 1.83i)6-s + (0.599 + 0.0788i)7-s + (−1.38 + 2.46i)8-s + (−2.99 − 0.188i)9-s + (−0.128 − 5.09i)10-s + (−2.82 − 5.72i)11-s + (3.42 − 0.517i)12-s + (0.527 + 1.55i)13-s + (0.579 + 0.628i)14-s + (−4.81 + 3.96i)15-s + (−3.73 + 1.41i)16-s + (−2.48 − 2.48i)17-s + ⋯
L(s)  = 1  + (0.768 + 0.640i)2-s + (0.0313 − 0.999i)3-s + (0.180 + 0.983i)4-s + (−1.06 − 1.21i)5-s + (0.663 − 0.747i)6-s + (0.226 + 0.0298i)7-s + (−0.491 + 0.871i)8-s + (−0.998 − 0.0626i)9-s + (−0.0406 − 1.61i)10-s + (−0.851 − 1.72i)11-s + (0.988 − 0.149i)12-s + (0.146 + 0.431i)13-s + (0.154 + 0.167i)14-s + (−1.24 + 1.02i)15-s + (−0.934 + 0.354i)16-s + (−0.602 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748815 - 1.03159i\)
\(L(\frac12)\) \(\approx\) \(0.748815 - 1.03159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.905i)T \)
3 \( 1 + (-0.0543 + 1.73i)T \)
good5 \( 1 + (2.37 + 2.70i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (-0.599 - 0.0788i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (2.82 + 5.72i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-0.527 - 1.55i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (2.48 + 2.48i)T + 17iT^{2} \)
19 \( 1 + (0.360 - 1.81i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (0.431 + 3.28i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-7.89 + 0.517i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (-2.49 + 1.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.97 + 9.94i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.516 - 3.92i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-6.04 + 2.98i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-1.06 + 0.285i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.409 + 0.612i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-6.59 - 7.52i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (0.876 + 13.3i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (-8.66 - 4.27i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-0.0723 - 0.174i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.42 - 5.85i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.21 - 4.53i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-7.35 - 6.44i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (4.84 - 2.00i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (10.5 + 6.11i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.05566964855482572505475510251, −8.902504251150914131846547280012, −8.336092140940260395592230330699, −7.935348185541480678475147249084, −6.83661288170322771135919923136, −5.82799340235337294438225768680, −4.98179626410864236124681797633, −3.89683417169870598571338892399, −2.64040205702376978229814912412, −0.53992991782422059685679361207, 2.44885070175414085670523174872, 3.30947250693932112893822668048, 4.35864452801593588607001733437, 4.94480622991340159004716613189, 6.34672599667392226455919562648, 7.30180978874355211066306494635, 8.354907706744254162200285225012, 9.763816077426176360744104138761, 10.38196887164345191463609853462, 10.89171343861373779145436190394

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