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Properties

Label 2-24e2-72.67-c2-0-4
Degree $2$
Conductor $576$
Sign $-0.958 - 0.283i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.06 + 2.80i)3-s + (−7.25 − 4.19i)5-s + (8.13 − 4.69i)7-s + (−6.74 − 5.96i)9-s + (6.44 + 11.1i)11-s + (14.4 + 8.32i)13-s + (19.4 − 15.9i)15-s − 6.33·17-s − 27.8·19-s + (4.53 + 27.8i)21-s + (−20.9 − 12.1i)23-s + (22.6 + 39.2i)25-s + (23.8 − 12.5i)27-s + (−34.4 + 19.8i)29-s + (16.4 + 9.50i)31-s + ⋯
L(s)  = 1  + (−0.354 + 0.935i)3-s + (−1.45 − 0.838i)5-s + (1.16 − 0.671i)7-s + (−0.749 − 0.662i)9-s + (0.585 + 1.01i)11-s + (1.10 + 0.640i)13-s + (1.29 − 1.06i)15-s − 0.372·17-s − 1.46·19-s + (0.215 + 1.32i)21-s + (−0.912 − 0.526i)23-s + (0.905 + 1.56i)25-s + (0.884 − 0.465i)27-s + (−1.18 + 0.685i)29-s + (0.531 + 0.306i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.283i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.958 - 0.283i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.958 - 0.283i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4006151882\)
\(L(\frac12)\) \(\approx\) \(0.4006151882\)
\(L(2)\) 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 \( 1 + (1.06 - 2.80i)T \)
good5 \( 1 + (7.25 + 4.19i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-8.13 + 4.69i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-6.44 - 11.1i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-14.4 - 8.32i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 6.33T + 289T^{2} \)
19 \( 1 + 27.8T + 361T^{2} \)
23 \( 1 + (20.9 + 12.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (34.4 - 19.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-16.4 - 9.50i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 1.38iT - 1.36e3T^{2} \)
41 \( 1 + (22.5 - 39.1i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (21.4 + 37.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5.86 - 3.38i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 30.9iT - 2.80e3T^{2} \)
59 \( 1 + (23.9 - 41.5i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (65.8 - 38.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.3 - 54.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 42.1iT - 5.04e3T^{2} \)
73 \( 1 + 123.T + 5.32e3T^{2} \)
79 \( 1 + (-37.2 + 21.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (27.3 + 47.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 61.8T + 7.92e3T^{2} \)
97 \( 1 + (-80.7 - 139. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.02848542954195972281666617639, −10.24278956406397033195404115197, −8.854103668216962624675333151115, −8.541747639113837581118539458207, −7.49479033763271641940665262174, −6.38451457798012634510625503089, −4.86974357595561859427296126528, −4.27481650105527550019108767430, −3.87328060962733028341073560686, −1.48999298927585183088793926909, 0.16666664336299067606269449781, 1.82093745552720602137357724812, 3.24173194022865123512085390228, 4.31876749327208340806862763561, 5.80538000579857935313224041556, 6.42841634288450031099273113731, 7.63408143667920022467333993145, 8.198747356689094029886207227633, 8.713695471476893885012526346008, 10.69447740063897740288326655574

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