Properties

Degree $2$
Conductor $288$
Sign $-0.450 - 0.892i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.360 − 1.96i)2-s + (−3.73 − 1.41i)4-s + (0.452 − 0.187i)5-s + (0.429 + 0.429i)7-s + (−4.14 + 6.84i)8-s + (−0.205 − 0.957i)10-s + (−17.3 + 7.18i)11-s + (−19.9 − 8.26i)13-s + (1.00 − 0.690i)14-s + (11.9 + 10.6i)16-s − 13.5i·17-s + (−3.45 + 8.34i)19-s + (−1.95 + 0.0583i)20-s + (7.86 + 36.6i)22-s + (16.8 − 16.8i)23-s + ⋯
L(s)  = 1  + (0.180 − 0.983i)2-s + (−0.934 − 0.354i)4-s + (0.0904 − 0.0374i)5-s + (0.0614 + 0.0614i)7-s + (−0.517 + 0.855i)8-s + (−0.0205 − 0.0957i)10-s + (−1.57 + 0.652i)11-s + (−1.53 − 0.635i)13-s + (0.0714 − 0.0493i)14-s + (0.747 + 0.663i)16-s − 0.799i·17-s + (−0.182 + 0.439i)19-s + (−0.0978 + 0.00291i)20-s + (0.357 + 1.66i)22-s + (0.734 − 0.734i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.450 - 0.892i$
Motivic weight: \(2\)
Character: $\chi_{288} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.450 - 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0542972 + 0.0882080i\)
\(L(\frac12)\) \(\approx\) \(0.0542972 + 0.0882080i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.360 + 1.96i)T \)
3 \( 1 \)
good5 \( 1 + (-0.452 + 0.187i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-0.429 - 0.429i)T + 49iT^{2} \)
11 \( 1 + (17.3 - 7.18i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (19.9 + 8.26i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 13.5iT - 289T^{2} \)
19 \( 1 + (3.45 - 8.34i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \)
29 \( 1 + (13.8 - 33.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 24.5iT - 961T^{2} \)
37 \( 1 + (-9.89 + 4.09i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (14.4 + 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (-17.8 + 7.39i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 43.6T + 2.20e3T^{2} \)
53 \( 1 + (28.0 + 67.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (1.70 + 4.10i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-3.53 + 8.53i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (0.300 + 0.124i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-29.0 - 29.0i)T + 5.04e3iT^{2} \)
73 \( 1 + (68.2 + 68.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 67.7T + 6.24e3T^{2} \)
83 \( 1 + (-16.4 + 39.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-45.3 + 45.3i)T - 7.92e3iT^{2} \)
97 \( 1 + 119.T + 9.40e3T^{2} \)
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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

−10.84561803484019065345785603105, −10.15706234054163803793583553643, −9.406035554209553514274975987817, −8.155360446005036063749015914189, −7.18662728333578723262113367171, −5.28523466366444528478710865140, −4.91452994475557497717468569889, −3.15742758487611074467392274882, −2.14117539301113189907769666658, −0.04493636373563733632066016347, 2.67428836301293772023865310850, 4.30012803126599931660816272222, 5.29778237443403994264361984947, 6.25718390431154819692501418033, 7.52491658271652992356251191647, 8.044483444856493176417445712541, 9.311244493153779174374864418120, 10.10422813343089612830862270837, 11.31955519324003737905437229303, 12.50779667721357984853400721127

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