Properties

Label 2-24e2-576.83-c1-0-26
Degree $2$
Conductor $576$
Sign $-0.744 - 0.667i$
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  + (−1.33 + 0.477i)2-s + (0.934 + 1.45i)3-s + (1.54 − 1.27i)4-s + (1.57 + 1.79i)5-s + (−1.94 − 1.49i)6-s + (−0.379 + 2.88i)7-s + (−1.44 + 2.43i)8-s + (−1.25 + 2.72i)9-s + (−2.95 − 1.63i)10-s + (4.15 − 2.05i)11-s + (3.29 + 1.06i)12-s + (−4.81 + 1.63i)13-s + (−0.872 − 4.01i)14-s + (−1.14 + 3.97i)15-s + (0.765 − 3.92i)16-s + (0.291 − 0.291i)17-s + ⋯
L(s)  = 1  + (−0.941 + 0.337i)2-s + (0.539 + 0.841i)3-s + (0.771 − 0.635i)4-s + (0.703 + 0.802i)5-s + (−0.792 − 0.610i)6-s + (−0.143 + 1.08i)7-s + (−0.511 + 0.859i)8-s + (−0.417 + 0.908i)9-s + (−0.933 − 0.517i)10-s + (1.25 − 0.618i)11-s + (0.951 + 0.306i)12-s + (−1.33 + 0.453i)13-s + (−0.233 − 1.07i)14-s + (−0.295 + 1.02i)15-s + (0.191 − 0.981i)16-s + (0.0706 − 0.0706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431679 + 1.12848i\)
\(L(\frac12)\) \(\approx\) \(0.431679 + 1.12848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.477i)T \)
3 \( 1 + (-0.934 - 1.45i)T \)
good5 \( 1 + (-1.57 - 1.79i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (0.379 - 2.88i)T + (-6.76 - 1.81i)T^{2} \)
11 \( 1 + (-4.15 + 2.05i)T + (6.69 - 8.72i)T^{2} \)
13 \( 1 + (4.81 - 1.63i)T + (10.3 - 7.91i)T^{2} \)
17 \( 1 + (-0.291 + 0.291i)T - 17iT^{2} \)
19 \( 1 + (1.13 - 5.69i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (1.09 + 8.30i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-5.91 + 0.387i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (1.09 + 1.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.87 + 1.56i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-8.57 + 1.12i)T + (39.6 - 10.6i)T^{2} \)
43 \( 1 + (-0.566 + 0.279i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (12.3 - 3.30i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.24 - 7.84i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-5.10 + 4.48i)T + (7.70 - 58.4i)T^{2} \)
61 \( 1 + (8.57 - 0.562i)T + (60.4 - 7.96i)T^{2} \)
67 \( 1 + (8.63 + 4.26i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-1.92 - 4.65i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.14 - 5.18i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (6.60 - 1.77i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.35 + 8.38i)T + (-10.8 - 82.2i)T^{2} \)
89 \( 1 + (-3.09 - 7.46i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (-12.4 - 7.16i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.67360427750457920890916924339, −9.936093257296701308381812347613, −9.361610507251153689822055074650, −8.668778321461185940123243596486, −7.75011637425404500879844634162, −6.35979112755041547818047515960, −6.00329210034058137957767768218, −4.55601489607437446976234207784, −2.87456908818606540938560086859, −2.16001270635721369943563794322, 0.885461843574364626738017889580, 1.87291534683510018993411434694, 3.20473489335444856119510307182, 4.59152096505884339701415205263, 6.23795975132553109209565264053, 7.15114007001228864567739191274, 7.63853851625367022693515490201, 8.811471645217195923070572302825, 9.539191149523002398089727928386, 9.931014697766859605057592697266

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