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Properties

Label 2-24e2-576.517-c1-0-42
Degree $2$
Conductor $576$
Sign $-0.311 - 0.950i$
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  + (1.03 + 0.968i)2-s + (1.63 + 0.565i)3-s + (0.123 + 1.99i)4-s + (−0.289 + 0.329i)5-s + (1.13 + 2.16i)6-s + (−0.132 + 0.0173i)7-s + (−1.80 + 2.17i)8-s + (2.35 + 1.85i)9-s + (−0.617 + 0.0597i)10-s + (−2.45 + 4.98i)11-s + (−0.926 + 3.33i)12-s + (1.43 − 4.22i)13-s + (−0.153 − 0.110i)14-s + (−0.660 + 0.376i)15-s + (−3.96 + 0.494i)16-s + (0.861 − 0.861i)17-s + ⋯
L(s)  = 1  + (0.728 + 0.684i)2-s + (0.945 + 0.326i)3-s + (0.0618 + 0.998i)4-s + (−0.129 + 0.147i)5-s + (0.464 + 0.885i)6-s + (−0.0499 + 0.00657i)7-s + (−0.638 + 0.769i)8-s + (0.786 + 0.617i)9-s + (−0.195 + 0.0188i)10-s + (−0.740 + 1.50i)11-s + (−0.267 + 0.963i)12-s + (0.398 − 1.17i)13-s + (−0.0409 − 0.0294i)14-s + (−0.170 + 0.0971i)15-s + (−0.992 + 0.123i)16-s + (0.208 − 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60220 + 2.21086i\)
\(L(\frac12)\) \(\approx\) \(1.60220 + 2.21086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 0.968i)T \)
3 \( 1 + (-1.63 - 0.565i)T \)
good5 \( 1 + (0.289 - 0.329i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (0.132 - 0.0173i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (2.45 - 4.98i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (-1.43 + 4.22i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-0.861 + 0.861i)T - 17iT^{2} \)
19 \( 1 + (1.11 + 5.60i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.478 + 3.63i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (-5.16 - 0.338i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-6.68 - 3.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.738 + 3.71i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.487 + 3.70i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (3.01 + 1.48i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (1.60 + 0.429i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.07 - 6.09i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-5.53 + 6.31i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (-0.930 + 14.1i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (5.53 - 2.72i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (4.00 - 9.67i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.92 + 7.05i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.29 + 8.57i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.38 + 1.21i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-2.54 - 1.05i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-11.1 + 6.41i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.85805003606879987876944901562, −10.08017277510493792803129134795, −9.001826925051078892936293300580, −8.155933841102874353033450349882, −7.42753939121692246334488785437, −6.66067516460470842041303856435, −5.12430282490945922392181837294, −4.57688046821167995966082993752, −3.26529997568170622201382345269, −2.46411268693879104250202156753, 1.26000445869096978983945324898, 2.62471273814118479162862428611, 3.56416463349735287632649920315, 4.46814445144355253683858042940, 5.89379256109075928067776373899, 6.60792238893516609995972359787, 8.078242460141441264313806252909, 8.616583831569384006631049397842, 9.759819747994830500749600167543, 10.43218195150917059375944284303

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