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Properties

Label 2-24e2-144.11-c1-0-20
Degree $2$
Conductor $576$
Sign $-0.999 + 0.0223i$
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  + (1.36 − 1.05i)3-s + (−2.78 − 0.746i)5-s + (−1.16 + 2.02i)7-s + (0.753 − 2.90i)9-s + (−5.53 + 1.48i)11-s + (−3.90 − 1.04i)13-s + (−4.60 + 1.93i)15-s − 6.45i·17-s + (1.50 + 1.50i)19-s + (0.543 + 4.00i)21-s + (0.0418 − 0.0241i)23-s + (2.87 + 1.66i)25-s + (−2.04 − 4.77i)27-s + (−5.08 + 1.36i)29-s + (1.65 − 0.952i)31-s + ⋯
L(s)  = 1  + (0.790 − 0.611i)3-s + (−1.24 − 0.333i)5-s + (−0.440 + 0.763i)7-s + (0.251 − 0.967i)9-s + (−1.67 + 0.447i)11-s + (−1.08 − 0.290i)13-s + (−1.19 + 0.498i)15-s − 1.56i·17-s + (0.345 + 0.345i)19-s + (0.118 + 0.873i)21-s + (0.00871 − 0.00503i)23-s + (0.575 + 0.332i)25-s + (−0.393 − 0.919i)27-s + (−0.944 + 0.253i)29-s + (0.296 − 0.171i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.999 + 0.0223i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.999 + 0.0223i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00425318 - 0.380065i\)
\(L(\frac12)\) \(\approx\) \(0.00425318 - 0.380065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 1.05i)T \)
good5 \( 1 + (2.78 + 0.746i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.53 - 1.48i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.90 + 1.04i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 6.45iT - 17T^{2} \)
19 \( 1 + (-1.50 - 1.50i)T + 19iT^{2} \)
23 \( 1 + (-0.0418 + 0.0241i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.08 - 1.36i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.65 + 0.952i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.489 - 0.489i)T + 37iT^{2} \)
41 \( 1 + (-0.0155 - 0.0269i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.01 - 3.80i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.0913 - 0.158i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.62 + 6.62i)T - 53iT^{2} \)
59 \( 1 + (1.11 - 4.15i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.71 + 6.39i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.216 - 0.808i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.04iT - 71T^{2} \)
73 \( 1 - 4.74iT - 73T^{2} \)
79 \( 1 + (7.29 + 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.20 + 11.9i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.85T + 89T^{2} \)
97 \( 1 + (-3.29 + 5.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.04442870071852801780193215269, −9.338295571833482155044109146478, −8.328970796979771786432822179119, −7.58474731374257243416459441644, −7.18601407879371049937746441828, −5.56755602508249550309059598116, −4.60099801966473485726038244347, −3.15843127990623414801656550594, −2.43996879485471958041590014568, −0.18038473306128355621084089626, 2.53230447749938576367590708537, 3.55856140483793324555533669376, 4.28573702018085106466659871887, 5.43634789330056489023119910838, 7.07610400595331827247276577262, 7.73701460895378086223236356572, 8.298113100843956164928437094624, 9.476170356083008240755148627483, 10.52581679810806909321005640004, 10.71185678718707818117204727221

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