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Properties

Label 2-24e2-72.59-c1-0-18
Degree $2$
Conductor $576$
Sign $-0.166 + 0.985i$
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  + (−1.46 − 0.925i)3-s + (2.06 − 3.57i)5-s + (0.287 − 0.165i)7-s + (1.28 + 2.71i)9-s + (3.62 − 2.09i)11-s + (4.27 + 2.46i)13-s + (−6.33 + 3.32i)15-s − 3.20i·17-s − 5.42·19-s + (−0.574 − 0.0231i)21-s + (−1.39 + 2.41i)23-s + (−6.02 − 10.4i)25-s + (0.625 − 5.15i)27-s + (−1.03 − 1.79i)29-s + (3.60 + 2.08i)31-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)3-s + (0.923 − 1.59i)5-s + (0.108 − 0.0627i)7-s + (0.428 + 0.903i)9-s + (1.09 − 0.630i)11-s + (1.18 + 0.684i)13-s + (−1.63 + 0.858i)15-s − 0.777i·17-s − 1.24·19-s + (−0.125 − 0.00504i)21-s + (−0.290 + 0.503i)23-s + (−1.20 − 2.08i)25-s + (0.120 − 0.992i)27-s + (−0.192 − 0.333i)29-s + (0.648 + 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.166 + 0.985i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.166 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888443 - 1.05144i\)
\(L(\frac12)\) \(\approx\) \(0.888443 - 1.05144i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 + 0.925i)T \)
good5 \( 1 + (-2.06 + 3.57i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.287 + 0.165i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.62 + 2.09i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.27 - 2.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 + 5.42T + 19T^{2} \)
23 \( 1 + (1.39 - 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.03 + 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.60 - 2.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.21iT - 37T^{2} \)
41 \( 1 + (-1.41 - 0.815i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.99 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.97 - 6.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 + (5.40 + 3.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.10 - 5.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.83 + 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.66T + 71T^{2} \)
73 \( 1 + 6.21T + 73T^{2} \)
79 \( 1 + (-0.719 + 0.415i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.96 + 3.44i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.14iT - 89T^{2} \)
97 \( 1 + (0.749 + 1.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.61737297496782346866471048839, −9.375174221356898544615386529806, −8.875087640359830056761342166270, −7.933288795646636251434389827571, −6.40472645831102892547627672183, −6.07653094026225511304442850203, −4.97490698432626618410252262538, −4.12361920890178033497343769703, −1.85220309556467714545094808509, −0.965350366042144559618922331079, 1.78287809078106832156911459651, 3.32607020953245687813493514918, 4.29828661438402307661312057278, 5.80767943205424640447959858442, 6.33664523332769240378474208194, 6.92137333006267593208137902682, 8.431727816735426750864403908648, 9.568403978834933604549861712104, 10.30258415034385019765109608629, 10.80118084474910184186856524097

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