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Properties

Label 2-24e2-144.13-c1-0-6
Degree $2$
Conductor $576$
Sign $-0.537 - 0.843i$
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  + (0.866 + 1.5i)3-s + (1 + 3.73i)5-s + (−0.633 + 0.366i)7-s + (−1.5 + 2.59i)9-s + (−2.86 − 0.767i)11-s + (6.09 − 1.63i)13-s + (−4.73 + 4.73i)15-s − 2.26·17-s + (0.633 + 0.633i)19-s + (−1.09 − 0.633i)21-s + (−1.09 − 0.633i)23-s + (−8.59 + 4.96i)25-s − 5.19·27-s + (0.633 − 2.36i)29-s + (3.73 − 6.46i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.447 + 1.66i)5-s + (−0.239 + 0.138i)7-s + (−0.5 + 0.866i)9-s + (−0.864 − 0.231i)11-s + (1.69 − 0.453i)13-s + (−1.22 + 1.22i)15-s − 0.550·17-s + (0.145 + 0.145i)19-s + (−0.239 − 0.138i)21-s + (−0.228 − 0.132i)23-s + (−1.71 + 0.992i)25-s − 1.00·27-s + (0.117 − 0.439i)29-s + (0.670 − 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.537 - 0.843i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.815528 + 1.48651i\)
\(L(\frac12)\) \(\approx\) \(0.815528 + 1.48651i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
good5 \( 1 + (-1 - 3.73i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.86 + 0.767i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-6.09 + 1.63i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + (-0.633 - 0.633i)T + 19iT^{2} \)
23 \( 1 + (1.09 + 0.633i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.633 + 2.36i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.73 + 6.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 1.26i)T - 37iT^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.23 - 0.330i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.83 - 8.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.535 - 0.535i)T - 53iT^{2} \)
59 \( 1 + (-1.33 - 4.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.803 + 3i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.23 + 1.40i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.366 - 1.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (4.13 + 7.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96431617928282344018660631302, −10.21213748778009593462252758042, −9.494864691044054263137169674058, −8.376275260043033615447225629169, −7.60241232225775011300711658489, −6.26950661889468743987492335439, −5.75477533926051155333888436969, −4.15782150827215961161591029308, −3.14999180499124618960374093457, −2.44121396100694338404737443707, 0.946072903074264022935386094781, 2.04753882268207566636267285774, 3.62963650697424264783565712871, 4.88034051281882439125679380690, 5.88060325453294493863199956584, 6.79201497109745060066968480096, 8.032516764123188955087856805595, 8.669279800795541501632997485123, 9.177005437003733392589509250171, 10.30741504937998300454803348063

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