Properties

Label 2-24e2-576.517-c1-0-41
Degree $2$
Conductor $576$
Sign $-0.276 + 0.961i$
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  + (0.325 − 1.37i)2-s + (−1.01 + 1.40i)3-s + (−1.78 − 0.894i)4-s + (−2.77 + 3.16i)5-s + (1.59 + 1.85i)6-s + (−1.04 + 0.137i)7-s + (−1.81 + 2.17i)8-s + (−0.930 − 2.85i)9-s + (3.45 + 4.85i)10-s + (2.72 − 5.52i)11-s + (3.07 − 1.59i)12-s + (−0.213 + 0.627i)13-s + (−0.149 + 1.47i)14-s + (−1.61 − 7.11i)15-s + (2.39 + 3.20i)16-s + (3.81 − 3.81i)17-s + ⋯
L(s)  = 1  + (0.229 − 0.973i)2-s + (−0.587 + 0.809i)3-s + (−0.894 − 0.447i)4-s + (−1.24 + 1.41i)5-s + (0.652 + 0.757i)6-s + (−0.393 + 0.0518i)7-s + (−0.641 + 0.767i)8-s + (−0.310 − 0.950i)9-s + (1.09 + 1.53i)10-s + (0.821 − 1.66i)11-s + (0.887 − 0.461i)12-s + (−0.0590 + 0.174i)13-s + (−0.0400 + 0.395i)14-s + (−0.417 − 1.83i)15-s + (0.599 + 0.800i)16-s + (0.924 − 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.276 + 0.961i$
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: \((2,\ 576,\ (\ :1/2),\ -0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360604 - 0.478834i\)
\(L(\frac12)\) \(\approx\) \(0.360604 - 0.478834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.325 + 1.37i)T \)
3 \( 1 + (1.01 - 1.40i)T \)
good5 \( 1 + (2.77 - 3.16i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (1.04 - 0.137i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (-2.72 + 5.52i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.213 - 0.627i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (-3.81 + 3.81i)T - 17iT^{2} \)
19 \( 1 + (-0.464 - 2.33i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.493 + 3.75i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (4.78 + 0.313i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-1.86 - 1.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.122 + 0.615i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.296 + 2.25i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-8.00 - 3.94i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (6.55 + 1.75i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.39 + 9.57i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (3.20 - 3.65i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.171 - 2.62i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (-2.49 + 1.23i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-3.32 + 8.02i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.27 + 7.91i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.439 - 1.64i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-10.0 + 8.78i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (13.7 + 5.70i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-6.87 + 3.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.72675405040254394402572678176, −9.929318974490043350496092785467, −9.018153376804595917083398804128, −7.945244178067342321199800386135, −6.56939505457351213747004294176, −5.81766508053727258926567742672, −4.43267313154297032296935562089, −3.38152865100253430441171502962, −3.18096723049956801043442348676, −0.40340281982039490913967112903, 1.23247821215521916557661876006, 3.76808040334177349739200413803, 4.64624454801841139941098559057, 5.43394487019761949334777457506, 6.58245325571752218741619186205, 7.55181636021630984145215880041, 7.87959058486997754046593265343, 8.997943285738905458643146863823, 9.768180363971149385817449577208, 11.40348750284453426977206964854

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