Properties

Label 2-24e2-36.7-c2-0-33
Degree $2$
Conductor $576$
Sign $-0.173 + 0.984i$
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  + (−1.5 + 2.59i)3-s + (2 + 3.46i)5-s + (3 + 1.73i)7-s + (−4.5 − 7.79i)9-s + (−10.5 − 6.06i)11-s + (−11 − 19.0i)13-s − 12·15-s − 11·17-s + 15.5i·19-s + (−9 + 5.19i)21-s + (−21 + 12.1i)23-s + (4.50 − 7.79i)25-s + 27·27-s + (17 − 29.4i)29-s + (−6 + 3.46i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.400 + 0.692i)5-s + (0.428 + 0.247i)7-s + (−0.5 − 0.866i)9-s + (−0.954 − 0.551i)11-s + (−0.846 − 1.46i)13-s − 0.800·15-s − 0.647·17-s + 0.820i·19-s + (−0.428 + 0.247i)21-s + (−0.913 + 0.527i)23-s + (0.180 − 0.311i)25-s + 27-s + (0.586 − 1.01i)29-s + (−0.193 + 0.111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 576,\ (\ :1),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3527382112\)
\(L(\frac12)\) \(\approx\) \(0.3527382112\)
\(L(2)\) 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 \( 1 + (1.5 - 2.59i)T \)
good5 \( 1 + (-2 - 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.5 + 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (11 + 19.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 11T + 289T^{2} \)
19 \( 1 - 15.5iT - 361T^{2} \)
23 \( 1 + (21 - 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17 + 29.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6 - 3.46i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 16T + 1.36e3T^{2} \)
41 \( 1 + (6.5 + 11.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (43.5 + 25.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (3 + 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 52T + 2.80e3T^{2} \)
59 \( 1 + (46.5 - 26.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (8 - 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-100.5 + 58.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + (24 + 13.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-30 - 17.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 2T + 7.92e3T^{2} \)
97 \( 1 + (-21.5 + 37.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.24941443540511059274347981873, −9.825301324173593974240884548717, −8.457355086581041208714180344989, −7.74693224643566380852781663556, −6.34211221004257398356933048986, −5.60190252630921423660167017704, −4.81899609213363227104480983324, −3.43144005074733749390539866625, −2.42981436620678005191324359228, −0.13549330151847848861141960238, 1.53745773506965180754326548574, 2.46291136828076195058768804096, 4.67680193322651878762482940399, 4.98493452305527111013193022697, 6.36876246564831212367085856633, 7.11407006396013270394615029892, 8.000455938676038075154183991720, 8.945896476952015821213181839251, 9.878827530601192824325410902550, 10.94700388788674152749362507194

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