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Properties

Label 2-24e2-36.7-c2-0-20
Degree $2$
Conductor $576$
Sign $0.477 - 0.878i$
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  + (2.32 + 1.89i)3-s + (−1.35 − 2.34i)5-s + (10.0 + 5.79i)7-s + (1.78 + 8.82i)9-s + (8.54 + 4.93i)11-s + (−0.296 − 0.513i)13-s + (1.31 − 8.03i)15-s − 8.87·17-s − 14.0i·19-s + (12.3 + 32.5i)21-s + (−18.2 + 10.5i)23-s + (8.82 − 15.2i)25-s + (−12.5 + 23.8i)27-s + (−10.1 + 17.6i)29-s + (14.3 − 8.27i)31-s + ⋯
L(s)  = 1  + (0.774 + 0.633i)3-s + (−0.271 − 0.469i)5-s + (1.43 + 0.828i)7-s + (0.198 + 0.980i)9-s + (0.777 + 0.448i)11-s + (−0.0227 − 0.0394i)13-s + (0.0873 − 0.535i)15-s − 0.522·17-s − 0.742i·19-s + (0.586 + 1.54i)21-s + (−0.794 + 0.458i)23-s + (0.352 − 0.611i)25-s + (−0.466 + 0.884i)27-s + (−0.350 + 0.607i)29-s + (0.462 − 0.266i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.477 - 0.878i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.657499011\)
\(L(\frac12)\) \(\approx\) \(2.657499011\)
\(L(2)\) 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 \( 1 + (-2.32 - 1.89i)T \)
good5 \( 1 + (1.35 + 2.34i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-10.0 - 5.79i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.54 - 4.93i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.296 + 0.513i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 8.87T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (18.2 - 10.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (10.1 - 17.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.3 + 8.27i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 40.6T + 1.36e3T^{2} \)
41 \( 1 + (-21.2 - 36.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.2 - 18.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.57 - 0.907i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 21.1T + 2.80e3T^{2} \)
59 \( 1 + (76.6 - 44.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (36.4 - 63.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38.3 + 22.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 + (8.30 + 4.79i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (73.6 + 42.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 64.7T + 7.92e3T^{2} \)
97 \( 1 + (3.59 - 6.22i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.72241491735402031083348735436, −9.494957803790088493502637843461, −8.919902356684152125391431077807, −8.196745177695694958531809436405, −7.43906411524077297514563906031, −5.91743409738907196140099077419, −4.66414994751678546962213216147, −4.35504479958194917498225628925, −2.71334086432629229148663573146, −1.60353702608658490375712314845, 1.04735645068354399112445599212, 2.19428988161764628380169291713, 3.65988937329863367337904406684, 4.40457200227977492170403364700, 5.99429005068464933184696391020, 7.01016461232720913636678155072, 7.75665991412012319452277132919, 8.373069343310311656306434648812, 9.327016901600974632460868710194, 10.49114732202202319799824540320

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