Properties

Label 2-24e2-144.61-c1-0-3
Degree $2$
Conductor $576$
Sign $-0.216 - 0.976i$
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  + 1.73i·3-s + (0.5 + 0.133i)5-s + (2.13 + 1.23i)7-s − 2.99·9-s + (−0.133 − 0.5i)11-s + (−1.23 + 4.59i)13-s + (−0.232 + 0.866i)15-s + 4·17-s + (3 + 3i)19-s + (−2.13 + 3.69i)21-s + (−0.401 + 0.232i)23-s + (−4.09 − 2.36i)25-s − 5.19i·27-s + (−3.23 + 0.866i)29-s + (−0.598 − 1.03i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.223 + 0.0599i)5-s + (0.806 + 0.465i)7-s − 0.999·9-s + (−0.0403 − 0.150i)11-s + (−0.341 + 1.27i)13-s + (−0.0599 + 0.223i)15-s + 0.970·17-s + (0.688 + 0.688i)19-s + (−0.465 + 0.806i)21-s + (−0.0838 + 0.0483i)23-s + (−0.819 − 0.473i)25-s − 0.999i·27-s + (−0.600 + 0.160i)29-s + (−0.107 − 0.186i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.216 - 0.976i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.216 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.940047 + 1.17127i\)
\(L(\frac12)\) \(\approx\) \(0.940047 + 1.17127i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
good5 \( 1 + (-0.5 - 0.133i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.13 - 1.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.133 + 0.5i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.23 - 4.59i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + (0.401 - 0.232i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.23 - 0.866i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.598 + 1.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.73 - 7.73i)T - 37iT^{2} \)
41 \( 1 + (-9.69 + 5.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.33 - 8.69i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.59 - 7.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.26 + 2.26i)T - 53iT^{2} \)
59 \( 1 + (5.59 + 1.5i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-14.4 + 3.86i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.330 + 1.23i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.7 + 3.16i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.01646988812635477903815080031, −9.910144076696271561552479803524, −9.438222684118417926292826791476, −8.433926509439944936160982320656, −7.61485953140342362977510081610, −6.17218802157949538224783852568, −5.33866764512297021622194394154, −4.45559449432927182013915747684, −3.34440003674585134994328785119, −1.91409019923414559038292049707, 0.896052327697105328122559191309, 2.24284944436441471978316693036, 3.56018545535216238145527784744, 5.20466998171835004367448205588, 5.74145650076230080048615771115, 7.22627305520648180321460997345, 7.60151306850621739276078406874, 8.483462672364116091131844436598, 9.599281429229323555327424371645, 10.59911754776883201813314409801

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