Properties

Degree $2$
Conductor $288$
Sign $-0.223 + 0.974i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.62 − 1.16i)2-s + (1.27 − 3.78i)4-s + (4.51 − 1.86i)5-s + (−3.85 − 3.85i)7-s + (−2.34 − 7.65i)8-s + (5.14 − 8.29i)10-s + (4.56 − 1.89i)11-s + (−5.58 − 2.31i)13-s + (−10.7 − 1.76i)14-s + (−12.7 − 9.70i)16-s + 25.0i·17-s + (6.43 − 15.5i)19-s + (−1.30 − 19.4i)20-s + (5.21 − 8.39i)22-s + (26.9 − 26.9i)23-s + ⋯
L(s)  = 1  + (0.812 − 0.583i)2-s + (0.319 − 0.947i)4-s + (0.902 − 0.373i)5-s + (−0.550 − 0.550i)7-s + (−0.292 − 0.956i)8-s + (0.514 − 0.829i)10-s + (0.414 − 0.171i)11-s + (−0.429 − 0.177i)13-s + (−0.768 − 0.126i)14-s + (−0.795 − 0.606i)16-s + 1.47i·17-s + (0.338 − 0.817i)19-s + (−0.0653 − 0.974i)20-s + (0.236 − 0.381i)22-s + (1.16 − 1.16i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.223 + 0.974i$
Motivic weight: \(2\)
Character: $\chi_{288} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.223 + 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.68796 - 2.11826i\)
\(L(\frac12)\) \(\approx\) \(1.68796 - 2.11826i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.62 + 1.16i)T \)
3 \( 1 \)
good5 \( 1 + (-4.51 + 1.86i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (3.85 + 3.85i)T + 49iT^{2} \)
11 \( 1 + (-4.56 + 1.89i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (5.58 + 2.31i)T + (119. + 119. i)T^{2} \)
17 \( 1 - 25.0iT - 289T^{2} \)
19 \( 1 + (-6.43 + 15.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-26.9 + 26.9i)T - 529iT^{2} \)
29 \( 1 + (-0.210 + 0.507i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 15.8iT - 961T^{2} \)
37 \( 1 + (2.18 - 0.905i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-31.1 - 31.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (-12.9 + 5.34i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 15.0T + 2.20e3T^{2} \)
53 \( 1 + (-15.4 - 37.2i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-14.7 - 35.5i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (15.4 - 37.3i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-61.3 - 25.4i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-51.7 - 51.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-64.9 - 64.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 38.1T + 6.24e3T^{2} \)
83 \( 1 + (15.9 - 38.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (23.7 - 23.7i)T - 7.92e3iT^{2} \)
97 \( 1 + 118.T + 9.40e3T^{2} \)
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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

−11.30525011976247621479568442356, −10.44298378277636292059162868329, −9.707357659248023744687655152262, −8.758036610433776197464648735381, −7.00633578147146011868000543335, −6.16606156679557560677990585159, −5.12973441001918692483326878887, −3.97354303058235170794130020540, −2.64093006073004918606087523324, −1.12099710739352697846721812290, 2.30075557559647296469424402042, 3.41781302413005880256131896531, 4.99338061545990409812005231017, 5.85318189584308070955823064129, 6.76841407769786654690057163533, 7.65158624478682096156973712462, 9.169353692646817278780491476964, 9.712350104494477759914793499054, 11.19784143606260832274966352971, 12.04529463429277568750132609152

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