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Properties

Label 2-24e2-36.7-c2-0-38
Degree $2$
Conductor $576$
Sign $0.180 + 0.983i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.82 + 1.01i)3-s + (−1.81 − 3.13i)5-s + (−1.59 − 0.920i)7-s + (6.93 + 5.73i)9-s + (−10.0 − 5.80i)11-s + (−6.43 − 11.1i)13-s + (−1.92 − 10.6i)15-s + 12.6·17-s − 25.6i·19-s + (−3.56 − 4.21i)21-s + (25.9 − 14.9i)23-s + (5.93 − 10.2i)25-s + (13.7 + 23.2i)27-s + (10.8 − 18.7i)29-s + (−52.4 + 30.2i)31-s + ⋯
L(s)  = 1  + (0.940 + 0.338i)3-s + (−0.362 − 0.627i)5-s + (−0.227 − 0.131i)7-s + (0.770 + 0.637i)9-s + (−0.914 − 0.528i)11-s + (−0.495 − 0.857i)13-s + (−0.128 − 0.713i)15-s + 0.742·17-s − 1.35i·19-s + (−0.169 − 0.200i)21-s + (1.12 − 0.650i)23-s + (0.237 − 0.411i)25-s + (0.509 + 0.860i)27-s + (0.372 − 0.645i)29-s + (−1.69 + 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.180 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.856000739\)
\(L(\frac12)\) \(\approx\) \(1.856000739\)
\(L(2)\) 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 \( 1 + (-2.82 - 1.01i)T \)
good5 \( 1 + (1.81 + 3.13i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (1.59 + 0.920i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.0 + 5.80i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (6.43 + 11.1i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 12.6T + 289T^{2} \)
19 \( 1 + 25.6iT - 361T^{2} \)
23 \( 1 + (-25.9 + 14.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-10.8 + 18.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (52.4 - 30.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 25.7T + 1.36e3T^{2} \)
41 \( 1 + (33.3 + 57.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (14.2 + 8.22i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-66.1 - 38.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 14.2T + 2.80e3T^{2} \)
59 \( 1 + (50.3 - 29.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-9.43 + 16.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (20.6 - 11.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 46.4iT - 5.04e3T^{2} \)
73 \( 1 - 49.3T + 5.32e3T^{2} \)
79 \( 1 + (52.4 + 30.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-86.2 - 49.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 154.T + 7.92e3T^{2} \)
97 \( 1 + (21.1 - 36.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.36926186104901298747485797126, −9.259069275030002674099217768876, −8.614798215643165947207954892796, −7.81057740463671780431598350415, −7.02475205935111610069327279442, −5.38221058343557593328416923078, −4.69618342072790520523775242123, −3.39394029256216867932376974227, −2.55949665659481282136843862279, −0.62596296210864487254388992637, 1.69002325943919251853949972081, 2.90662138518172555255369149993, 3.73528423406373206478794821898, 5.07426335673799829657511115113, 6.41513142703201550938483863374, 7.48829142758165313892244279358, 7.71268826235266666635151282505, 9.036447252398391472616214173520, 9.717316027509657715273205491469, 10.57647901535463316521549298094

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