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Properties

Label 2-24e2-144.11-c1-0-15
Degree $2$
Conductor $576$
Sign $0.590 + 0.807i$
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  + (1.65 − 0.517i)3-s + (−1.94 − 0.521i)5-s + (0.322 − 0.558i)7-s + (2.46 − 1.71i)9-s + (5.59 − 1.49i)11-s + (−1.97 − 0.530i)13-s + (−3.48 + 0.144i)15-s + 3.05i·17-s + (−4.11 − 4.11i)19-s + (0.243 − 1.08i)21-s + (7.05 − 4.07i)23-s + (−0.820 − 0.473i)25-s + (3.18 − 4.10i)27-s + (4.49 − 1.20i)29-s + (−0.147 + 0.0854i)31-s + ⋯
L(s)  = 1  + (0.954 − 0.298i)3-s + (−0.869 − 0.233i)5-s + (0.121 − 0.210i)7-s + (0.821 − 0.570i)9-s + (1.68 − 0.451i)11-s + (−0.548 − 0.147i)13-s + (−0.899 + 0.0373i)15-s + 0.741i·17-s + (−0.943 − 0.943i)19-s + (0.0532 − 0.237i)21-s + (1.47 − 0.849i)23-s + (−0.164 − 0.0947i)25-s + (0.613 − 0.789i)27-s + (0.834 − 0.223i)29-s + (−0.0265 + 0.0153i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.590 + 0.807i$
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: \((2,\ 576,\ (\ :1/2),\ 0.590 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65707 - 0.840995i\)
\(L(\frac12)\) \(\approx\) \(1.65707 - 0.840995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 0.517i)T \)
good5 \( 1 + (1.94 + 0.521i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.322 + 0.558i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.59 + 1.49i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.97 + 0.530i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.05iT - 17T^{2} \)
19 \( 1 + (4.11 + 4.11i)T + 19iT^{2} \)
23 \( 1 + (-7.05 + 4.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.49 + 1.20i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.147 - 0.0854i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.65 + 2.65i)T + 37iT^{2} \)
41 \( 1 + (-0.983 - 1.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.220 - 0.821i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.02 - 6.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.25 - 3.25i)T - 53iT^{2} \)
59 \( 1 + (0.135 - 0.506i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.805 + 3.00i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.583 - 2.17i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 - 9.33iT - 73T^{2} \)
79 \( 1 + (-9.44 - 5.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.31 + 12.3i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.86T + 89T^{2} \)
97 \( 1 + (7.33 - 12.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64887245883658142820380561927, −9.409249925318538875418402353069, −8.742559097852334141098644656265, −8.103442070215545020858985102516, −7.04262236289563168228169913566, −6.39823396038645756401752004959, −4.54427955880249621176351204308, −3.92989457308224723943434605224, −2.72720783576017339987664257003, −1.09140972907460208129201365564, 1.75444676656493494542736435691, 3.23721476218949756612339241234, 4.03365746742015078947385460113, 4.97642550007983874241016513805, 6.68387247532527555261778714504, 7.31679068598302660091625683265, 8.294509257947272676919102310846, 9.103310677347711927702075439980, 9.757943507146830675289941371265, 10.83195701098779383021110336204

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