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Properties

Label 2-24e2-144.83-c1-0-18
Degree $2$
Conductor $576$
Sign $-0.689 + 0.724i$
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  + (−1.53 − 0.800i)3-s + (1.00 − 3.73i)5-s + (1.68 − 2.91i)7-s + (1.71 + 2.45i)9-s + (−0.0566 − 0.211i)11-s + (0.727 − 2.71i)13-s + (−4.52 + 4.93i)15-s + 4.23i·17-s + (1.12 − 1.12i)19-s + (−4.91 + 3.13i)21-s + (−3.33 + 1.92i)23-s + (−8.61 − 4.97i)25-s + (−0.675 − 5.15i)27-s + (0.545 + 2.03i)29-s + (7.21 − 4.16i)31-s + ⋯
L(s)  = 1  + (−0.886 − 0.461i)3-s + (0.447 − 1.67i)5-s + (0.635 − 1.10i)7-s + (0.573 + 0.819i)9-s + (−0.0170 − 0.0637i)11-s + (0.201 − 0.753i)13-s + (−1.16 + 1.27i)15-s + 1.02i·17-s + (0.257 − 0.257i)19-s + (−1.07 + 0.683i)21-s + (−0.695 + 0.401i)23-s + (−1.72 − 0.994i)25-s + (−0.129 − 0.991i)27-s + (0.101 + 0.378i)29-s + (1.29 − 0.747i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.689 + 0.724i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462121 - 1.07722i\)
\(L(\frac12)\) \(\approx\) \(0.462121 - 1.07722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 + 0.800i)T \)
good5 \( 1 + (-1.00 + 3.73i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.68 + 2.91i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0566 + 0.211i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.727 + 2.71i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 - 4.23iT - 17T^{2} \)
19 \( 1 + (-1.12 + 1.12i)T - 19iT^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.545 - 2.03i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-7.21 + 4.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.66 - 2.66i)T - 37iT^{2} \)
41 \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.68 - 1.25i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.34 - 4.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.58 + 7.58i)T + 53iT^{2} \)
59 \( 1 + (-5.34 - 1.43i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-8.69 + 2.33i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.17 - 1.38i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.53iT - 71T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 + (-4.98 - 2.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.50 - 1.20i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.96T + 89T^{2} \)
97 \( 1 + (7.63 - 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.39473063880964977743238146885, −9.731095761285006602072860529208, −8.272339524780264791377201832473, −7.998778668452830527959729770563, −6.66710654462052787997234405633, −5.62651272818382575634518514225, −4.92622647938653626248143776311, −4.05203314015021609413406638205, −1.68084200853177046702228353614, −0.78854529992998692535181939911, 2.07001383320480876740653475484, 3.23758488092676973866450265864, 4.65146763370519630931402037333, 5.66907207861340597963235484692, 6.44217155707491034572670946957, 7.14570347544044038470557372529, 8.490391003936630940294606228580, 9.654532410107705567801758436545, 10.17928881976023891763048216145, 11.18349595807511987275543390571

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