Properties

Label 2-24e2-576.133-c1-0-92
Degree $2$
Conductor $576$
Sign $-0.805 - 0.592i$
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  + (0.780 − 1.17i)2-s + (−0.286 − 1.70i)3-s + (−0.781 − 1.84i)4-s + (0.396 + 1.16i)5-s + (−2.23 − 0.995i)6-s + (−2.38 − 3.10i)7-s + (−2.78 − 0.516i)8-s + (−2.83 + 0.979i)9-s + (1.68 + 0.444i)10-s + (−2.37 + 0.155i)11-s + (−2.92 + 1.86i)12-s + (2.84 + 3.24i)13-s + (−5.52 + 0.385i)14-s + (1.88 − 1.01i)15-s + (−2.77 + 2.87i)16-s + (1.95 − 1.95i)17-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + (−0.165 − 0.986i)3-s + (−0.390 − 0.920i)4-s + (0.177 + 0.522i)5-s + (−0.913 − 0.406i)6-s + (−0.900 − 1.17i)7-s + (−0.983 − 0.182i)8-s + (−0.945 + 0.326i)9-s + (0.533 + 0.140i)10-s + (−0.716 + 0.0469i)11-s + (−0.843 + 0.537i)12-s + (0.789 + 0.900i)13-s + (−1.47 + 0.103i)14-s + (0.486 − 0.261i)15-s + (−0.694 + 0.719i)16-s + (0.475 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332270 + 1.01189i\)
\(L(\frac12)\) \(\approx\) \(0.332270 + 1.01189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.780 + 1.17i)T \)
3 \( 1 + (0.286 + 1.70i)T \)
good5 \( 1 + (-0.396 - 1.16i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (2.38 + 3.10i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (2.37 - 0.155i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (-2.84 - 3.24i)T + (-1.69 + 12.8i)T^{2} \)
17 \( 1 + (-1.95 + 1.95i)T - 17iT^{2} \)
19 \( 1 + (0.121 + 0.611i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (6.47 + 4.97i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (0.0660 + 0.133i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (2.00 - 1.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.20 + 6.07i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.0233 - 0.0179i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (0.795 + 12.1i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (-1.87 - 6.99i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.33 + 6.49i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-1.83 - 5.40i)T + (-46.8 + 35.9i)T^{2} \)
61 \( 1 + (-3.96 + 1.95i)T + (37.1 - 48.3i)T^{2} \)
67 \( 1 + (-0.474 + 7.24i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (3.74 - 9.02i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.63 + 6.35i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.76 + 0.471i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.77 - 2.63i)T + (65.8 + 50.5i)T^{2} \)
89 \( 1 + (0.129 + 0.0535i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (15.2 + 8.82i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.54050159373983897603949514452, −9.629658949402225115401423681208, −8.460451199538687743162779408233, −7.17738523613852911650259526833, −6.53932680987902822614806778931, −5.69480519995770181478142190764, −4.26172234946801359655058014207, −3.19433392999721375496706404662, −2.08421676270198220779595346833, −0.49353451483330027048869007009, 2.90566453456666056852703326773, 3.70427319639115612022304101170, 5.04333964383728215695153668262, 5.73820488004788455631345238203, 6.20045054289721417390565461735, 7.903317375824248084971941881536, 8.562063974134717158840284572593, 9.404828139902793766031146859348, 10.12225617070461432120379769774, 11.36903537817086916261553309151

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