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Properties

Label 2-24e2-36.7-c2-0-25
Degree $2$
Conductor $576$
Sign $-0.493 - 0.869i$
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  + (−0.531 + 2.95i)3-s + (3.31 + 5.73i)5-s + (8.46 + 4.88i)7-s + (−8.43 − 3.13i)9-s + (10.0 + 5.80i)11-s + (8.93 + 15.4i)13-s + (−18.6 + 6.72i)15-s + 2.37·17-s − 14.0i·19-s + (−18.9 + 22.4i)21-s + (35.9 − 20.7i)23-s + (−9.43 + 16.3i)25-s + (13.7 − 23.2i)27-s + (5.68 − 9.85i)29-s + (18.0 − 10.4i)31-s + ⋯
L(s)  = 1  + (−0.177 + 0.984i)3-s + (0.662 + 1.14i)5-s + (1.20 + 0.698i)7-s + (−0.937 − 0.348i)9-s + (0.914 + 0.528i)11-s + (0.687 + 1.19i)13-s + (−1.24 + 0.448i)15-s + 0.139·17-s − 0.738i·19-s + (−0.901 + 1.06i)21-s + (1.56 − 0.902i)23-s + (−0.377 + 0.653i)25-s + (0.509 − 0.860i)27-s + (0.196 − 0.339i)29-s + (0.581 − 0.335i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.493 - 0.869i$
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: \((2,\ 576,\ (\ :1),\ -0.493 - 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.338349314\)
\(L(\frac12)\) \(\approx\) \(2.338349314\)
\(L(2)\) 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 \( 1 + (0.531 - 2.95i)T \)
good5 \( 1 + (-3.31 - 5.73i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-8.46 - 4.88i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.0 - 5.80i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.93 - 15.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 2.37T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (-35.9 + 20.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-5.68 + 9.85i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-18.0 + 10.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 35.7T + 1.36e3T^{2} \)
41 \( 1 + (2.62 + 4.55i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (54.4 + 31.4i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (4.28 + 2.47i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 75.7T + 2.80e3T^{2} \)
59 \( 1 + (-50.3 + 29.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.93 - 10.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (20.6 - 11.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 46.4iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + (-18.0 - 10.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (24.4 + 14.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 73.0T + 7.92e3T^{2} \)
97 \( 1 + (-71.1 + 123. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.90482383465933836681775767088, −9.986223370637125838423110456140, −9.079771199983861749375640171881, −8.533443486534264812507055089841, −6.91783056940080258015731289080, −6.32332057339516771634009570882, −5.12715743038004551191997786273, −4.33087399157782958090700234332, −2.99548646711911654504632740385, −1.82345522712763143334757771238, 1.17195720124470106972130620136, 1.36138890999409085298049751888, 3.34766787798366273153207278665, 4.90908384861233065270763062662, 5.53165298902039749897346061061, 6.55979837685074925668198008853, 7.71513723025226469343008240955, 8.387971485175045883228331960640, 9.049165126395039167997205581724, 10.38000284545502672154521449233

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