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Properties

Label 2-24e2-16.3-c2-0-1
Degree $2$
Conductor $576$
Sign $-0.253 - 0.967i$
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  + (−0.227 + 0.227i)5-s − 3.90·7-s + (−2.21 − 2.21i)11-s + (5.08 + 5.08i)13-s + 18.8·17-s + (−11.7 + 11.7i)19-s − 35.4·23-s + 24.8i·25-s + (21.2 + 21.2i)29-s + 35.9i·31-s + (0.888 − 0.888i)35-s + (−34.4 + 34.4i)37-s − 44.1i·41-s + (28.3 + 28.3i)43-s − 32.8i·47-s + ⋯
L(s)  = 1  + (−0.0455 + 0.0455i)5-s − 0.557·7-s + (−0.200 − 0.200i)11-s + (0.391 + 0.391i)13-s + 1.10·17-s + (−0.619 + 0.619i)19-s − 1.54·23-s + 0.995i·25-s + (0.732 + 0.732i)29-s + 1.16i·31-s + (0.0253 − 0.0253i)35-s + (−0.930 + 0.930i)37-s − 1.07i·41-s + (0.658 + 0.658i)43-s − 0.699i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.253 - 0.967i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.253 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.056175917\)
\(L(\frac12)\) \(\approx\) \(1.056175917\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (0.227 - 0.227i)T - 25iT^{2} \)
7 \( 1 + 3.90T + 49T^{2} \)
11 \( 1 + (2.21 + 2.21i)T + 121iT^{2} \)
13 \( 1 + (-5.08 - 5.08i)T + 169iT^{2} \)
17 \( 1 - 18.8T + 289T^{2} \)
19 \( 1 + (11.7 - 11.7i)T - 361iT^{2} \)
23 \( 1 + 35.4T + 529T^{2} \)
29 \( 1 + (-21.2 - 21.2i)T + 841iT^{2} \)
31 \( 1 - 35.9iT - 961T^{2} \)
37 \( 1 + (34.4 - 34.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 44.1iT - 1.68e3T^{2} \)
43 \( 1 + (-28.3 - 28.3i)T + 1.84e3iT^{2} \)
47 \( 1 + 32.8iT - 2.20e3T^{2} \)
53 \( 1 + (42.1 - 42.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-66.9 - 66.9i)T + 3.48e3iT^{2} \)
61 \( 1 + (-17.3 - 17.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (39.6 - 39.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 63.0T + 5.04e3T^{2} \)
73 \( 1 - 75.4iT - 5.32e3T^{2} \)
79 \( 1 - 59.1iT - 6.24e3T^{2} \)
83 \( 1 + (-71.2 + 71.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 150. iT - 7.92e3T^{2} \)
97 \( 1 - 51.5T + 9.40e3T^{2} \)
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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

−10.53505282898659371231882325541, −10.08117613373183207552261807127, −8.969765013101772580249790014627, −8.187840423882741226772709813569, −7.16137730561189991746968002326, −6.21356979590206702378331521707, −5.34515054536511608125434514850, −3.98754373610597471354574149903, −3.07607622627052672617148991061, −1.49185190653948740492051983381, 0.40451630129624684223463397199, 2.22086491824656994168152946631, 3.47613519679248933149878366422, 4.53412819027063429292859788654, 5.80901120471553188584631665049, 6.49180127043518256069016729522, 7.74777400395102536263627238733, 8.353543757608780470784780292050, 9.593447704571986572842519938151, 10.13034895064413808240324383664

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