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Properties

Label 2-24e2-144.131-c1-0-6
Degree $2$
Conductor $576$
Sign $-0.0772 - 0.997i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.412 + 1.68i)3-s + (0.289 − 0.0776i)5-s + (0.374 + 0.647i)7-s + (−2.65 + 1.38i)9-s + (2.23 + 0.599i)11-s + (1.60 − 0.429i)13-s + (0.250 + 0.455i)15-s + 6.74i·17-s + (−0.621 + 0.621i)19-s + (−0.935 + 0.896i)21-s + (6.06 + 3.49i)23-s + (−4.25 + 2.45i)25-s + (−3.43 − 3.90i)27-s + (−5.44 − 1.45i)29-s + (−3.13 − 1.81i)31-s + ⋯
L(s)  = 1  + (0.238 + 0.971i)3-s + (0.129 − 0.0347i)5-s + (0.141 + 0.244i)7-s + (−0.886 + 0.462i)9-s + (0.674 + 0.180i)11-s + (0.444 − 0.119i)13-s + (0.0646 + 0.117i)15-s + 1.63i·17-s + (−0.142 + 0.142i)19-s + (−0.204 + 0.195i)21-s + (1.26 + 0.729i)23-s + (−0.850 + 0.490i)25-s + (−0.660 − 0.750i)27-s + (−1.01 − 0.270i)29-s + (−0.563 − 0.325i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0772 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0772 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.0772 - 0.997i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.0772 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06690 + 1.15279i\)
\(L(\frac12)\) \(\approx\) \(1.06690 + 1.15279i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.412 - 1.68i)T \)
good5 \( 1 + (-0.289 + 0.0776i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.374 - 0.647i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.23 - 0.599i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.60 + 0.429i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 6.74iT - 17T^{2} \)
19 \( 1 + (0.621 - 0.621i)T - 19iT^{2} \)
23 \( 1 + (-6.06 - 3.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.44 + 1.45i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.13 + 1.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.74 + 6.74i)T - 37iT^{2} \)
41 \( 1 + (1.39 - 2.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.89 - 7.08i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.307 + 0.531i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.68 - 2.68i)T + 53iT^{2} \)
59 \( 1 + (-0.00225 - 0.00841i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.72 + 10.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.35 + 8.78i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 - 8.17iT - 73T^{2} \)
79 \( 1 + (-7.67 + 4.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.353 + 1.32i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + (4.62 + 8.00i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06119057341140393237219450229, −9.944573039781263799104902736045, −9.277461939121020705191019482648, −8.512464803339935983802723876405, −7.57326615507610415850804651449, −6.15362293948868480687580083536, −5.44241966276420834349797619488, −4.17329865614594276216468584446, −3.45820944149513305928924645732, −1.87090731128325038297540170031, 0.923375666365451184935356125144, 2.36658352048752700446682554088, 3.56039575267356674612623326092, 4.97104390861369830075168081838, 6.14277038086260728096768099621, 6.96299501627198871238331399657, 7.67024163881181676494321281153, 8.819125068500814236011351616377, 9.311632652057292089671299891962, 10.64520953245271601335179639944

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