Properties

Label 2-24e2-144.11-c1-0-13
Degree $2$
Conductor $576$
Sign $0.966 - 0.256i$
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  + (1.71 + 0.256i)3-s + (2.03 + 0.546i)5-s + (0.0638 − 0.110i)7-s + (2.86 + 0.878i)9-s + (0.678 − 0.181i)11-s + (−1.84 − 0.493i)13-s + (3.35 + 1.45i)15-s − 4.32i·17-s + (3.97 + 3.97i)19-s + (0.137 − 0.172i)21-s + (−6.81 + 3.93i)23-s + (−0.475 − 0.274i)25-s + (4.68 + 2.23i)27-s + (−0.926 + 0.248i)29-s + (4.91 − 2.83i)31-s + ⋯
L(s)  = 1  + (0.988 + 0.148i)3-s + (0.911 + 0.244i)5-s + (0.0241 − 0.0417i)7-s + (0.956 + 0.292i)9-s + (0.204 − 0.0548i)11-s + (−0.511 − 0.136i)13-s + (0.865 + 0.376i)15-s − 1.04i·17-s + (0.912 + 0.912i)19-s + (0.0300 − 0.0377i)21-s + (−1.42 + 0.820i)23-s + (−0.0951 − 0.0549i)25-s + (0.902 + 0.431i)27-s + (−0.172 + 0.0461i)29-s + (0.882 − 0.509i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33121 + 0.303909i\)
\(L(\frac12)\) \(\approx\) \(2.33121 + 0.303909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.71 - 0.256i)T \)
good5 \( 1 + (-2.03 - 0.546i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.0638 + 0.110i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.678 + 0.181i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.84 + 0.493i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 4.32iT - 17T^{2} \)
19 \( 1 + (-3.97 - 3.97i)T + 19iT^{2} \)
23 \( 1 + (6.81 - 3.93i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.926 - 0.248i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-4.91 + 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.64 + 6.64i)T + 37iT^{2} \)
41 \( 1 + (-4.61 - 7.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.91 + 10.8i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.92 - 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.00259 + 0.00259i)T - 53iT^{2} \)
59 \( 1 + (-1.09 + 4.09i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.19 + 4.44i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.538 + 2.01i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.80iT - 71T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 + (3.00 + 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.394 - 1.47i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (-3.31 + 5.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.40168124440384826425917050231, −9.703559059603311020742969108587, −9.315267775988097517654816295044, −8.013906308858005991093591581958, −7.41277733901414409304567001888, −6.21363832714870385874484753659, −5.19812113208024162863668983398, −3.92477948467387070033371527375, −2.80663400014198528695533952460, −1.73977448419219812832593110855, 1.58663647940419112244083818588, 2.61226207462272006502145966036, 3.90924662496505387988769343607, 5.05597664863651533305829264738, 6.25024196452093592966929641417, 7.12978265239378471998107482356, 8.237413741829935148639477870696, 8.874898594024618090447688546041, 9.890640285372077192265486855492, 10.20587054767362015873555949378

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