Properties

Degree $2$
Conductor $288$
Sign $0.697 - 0.716i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 + 0.229i)2-s + (3.89 + 0.912i)4-s + (−4.18 + 1.73i)5-s + (3.93 + 3.93i)7-s + (7.52 + 2.70i)8-s + (−8.72 + 2.48i)10-s + (14.2 − 5.89i)11-s + (0.454 + 0.188i)13-s + (6.90 + 8.71i)14-s + (14.3 + 7.11i)16-s + 26.5i·17-s + (−7.25 + 17.5i)19-s + (−17.8 + 2.93i)20-s + (29.6 − 8.44i)22-s + (−0.775 + 0.775i)23-s + ⋯
L(s)  = 1  + (0.993 + 0.114i)2-s + (0.973 + 0.228i)4-s + (−0.837 + 0.346i)5-s + (0.561 + 0.561i)7-s + (0.940 + 0.338i)8-s + (−0.872 + 0.248i)10-s + (1.29 − 0.536i)11-s + (0.0349 + 0.0144i)13-s + (0.493 + 0.622i)14-s + (0.895 + 0.444i)16-s + 1.56i·17-s + (−0.381 + 0.921i)19-s + (−0.894 + 0.146i)20-s + (1.34 − 0.383i)22-s + (−0.0337 + 0.0337i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.697 - 0.716i$
Motivic weight: \(2\)
Character: $\chi_{288} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.697 - 0.716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.69693 + 1.13831i\)
\(L(\frac12)\) \(\approx\) \(2.69693 + 1.13831i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 - 0.229i)T \)
3 \( 1 \)
good5 \( 1 + (4.18 - 1.73i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-3.93 - 3.93i)T + 49iT^{2} \)
11 \( 1 + (-14.2 + 5.89i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-0.454 - 0.188i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 26.5iT - 289T^{2} \)
19 \( 1 + (7.25 - 17.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (0.775 - 0.775i)T - 529iT^{2} \)
29 \( 1 + (-17.9 + 43.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 39.6iT - 961T^{2} \)
37 \( 1 + (-36.4 + 15.1i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (38.9 + 38.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (14.2 - 5.91i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 62.1T + 2.20e3T^{2} \)
53 \( 1 + (11.4 + 27.7i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-5.30 - 12.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-14.1 + 34.1i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-26.1 - 10.8i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (17.7 + 17.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-12.8 - 12.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 144.T + 6.24e3T^{2} \)
83 \( 1 + (-10.9 + 26.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-5.92 + 5.92i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.9T + 9.40e3T^{2} \)
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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

−11.61307764446608533810320479076, −11.34714612832214255458931843065, −10.07196071950646114065484343253, −8.454253338227126782388969826500, −7.85287391237840665268116198866, −6.48962201874057862459821145219, −5.80095479129065313083092091369, −4.23764589073776621616117818124, −3.60409649774622693194151094958, −1.89158885002274609009535184186, 1.26313060683646985789981225109, 3.12899978904042590444170354637, 4.42295451707215493530238367249, 4.89315192841635336410853924930, 6.66037788163393052325940791388, 7.24595142181438352945088981177, 8.459804917932411075727043261483, 9.713547282150187651876819394366, 10.96238833073752866178439450847, 11.66651112842830029826611582542

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