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Properties

Label 2-24e2-576.11-c1-0-71
Degree $2$
Conductor $576$
Sign $-0.990 + 0.138i$
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  + (−1.37 + 0.318i)2-s + (−0.139 − 1.72i)3-s + (1.79 − 0.877i)4-s + (−0.245 − 0.121i)5-s + (0.742 + 2.33i)6-s + (0.961 + 1.25i)7-s + (−2.19 + 1.78i)8-s + (−2.96 + 0.481i)9-s + (0.376 + 0.0886i)10-s + (−2.50 − 2.19i)11-s + (−1.76 − 2.98i)12-s + (0.468 + 0.0307i)13-s + (−1.72 − 1.41i)14-s + (−0.174 + 0.440i)15-s + (2.45 − 3.15i)16-s + (−4.48 − 4.48i)17-s + ⋯
L(s)  = 1  + (−0.974 + 0.225i)2-s + (−0.0804 − 0.996i)3-s + (0.898 − 0.438i)4-s + (−0.109 − 0.0541i)5-s + (0.302 + 0.953i)6-s + (0.363 + 0.473i)7-s + (−0.776 + 0.630i)8-s + (−0.987 + 0.160i)9-s + (0.119 + 0.0280i)10-s + (−0.755 − 0.662i)11-s + (−0.509 − 0.860i)12-s + (0.130 + 0.00852i)13-s + (−0.460 − 0.379i)14-s + (−0.0451 + 0.113i)15-s + (0.614 − 0.788i)16-s + (−1.08 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.990 + 0.138i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0235919 - 0.338722i\)
\(L(\frac12)\) \(\approx\) \(0.0235919 - 0.338722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.318i)T \)
3 \( 1 + (0.139 + 1.72i)T \)
good5 \( 1 + (0.245 + 0.121i)T + (3.04 + 3.96i)T^{2} \)
7 \( 1 + (-0.961 - 1.25i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (2.50 + 2.19i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (-0.468 - 0.0307i)T + (12.8 + 1.69i)T^{2} \)
17 \( 1 + (4.48 + 4.48i)T + 17iT^{2} \)
19 \( 1 + (1.86 - 2.78i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-3.20 + 4.17i)T + (-5.95 - 22.2i)T^{2} \)
29 \( 1 + (1.39 - 4.11i)T + (-23.0 - 17.6i)T^{2} \)
31 \( 1 + (2.61 + 4.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.67 - 5.12i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (4.44 + 3.41i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (-4.90 - 4.30i)T + (5.61 + 42.6i)T^{2} \)
47 \( 1 + (-0.164 - 0.612i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.57 + 1.30i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (6.50 - 13.1i)T + (-35.9 - 46.8i)T^{2} \)
61 \( 1 + (-4.18 + 12.3i)T + (-48.3 - 37.1i)T^{2} \)
67 \( 1 + (-1.62 + 1.42i)T + (8.74 - 66.4i)T^{2} \)
71 \( 1 + (4.92 + 2.03i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (8.43 - 3.49i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.0695 + 0.259i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-9.48 + 4.67i)T + (50.5 - 65.8i)T^{2} \)
89 \( 1 + (3.52 + 1.46i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-1.95 - 1.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.41432348881250508769728450289, −9.047105055457442212578107003744, −8.505537670980923783572149324575, −7.75637683901117970840080728943, −6.84602357265009372277719232420, −6.01852358711118902658995596955, −5.05983134200117030552120021535, −2.89690563285498564319452930170, −1.91352687182706468810150627641, −0.24547394161174945507692961553, 1.98453095945181097323662777123, 3.39340152379902012711422778287, 4.43109612907792891813764209239, 5.63835734773970553381308665871, 6.89241248330888203644474567483, 7.78327192399919920893055162400, 8.738166762687043248501658740814, 9.393823913039917201503241322804, 10.40396212956563976436690566126, 10.84886879591444929314864122277

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