Properties

Degree $2$
Conductor $288$
Sign $0.196 - 0.980i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 1.17i)2-s + (1.23 + 3.80i)4-s + (−2.28 + 0.948i)5-s + (−6.37 − 6.37i)7-s + (2.48 − 7.60i)8-s + (4.81 + 1.15i)10-s + (1.79 − 0.744i)11-s + (16.7 + 6.91i)13-s + (2.81 + 17.8i)14-s + (−12.9 + 9.38i)16-s − 6.19i·17-s + (−8.50 + 20.5i)19-s + (−6.42 − 7.53i)20-s + (−3.78 − 0.909i)22-s + (−23.6 + 23.6i)23-s + ⋯
L(s)  = 1  + (−0.808 − 0.588i)2-s + (0.308 + 0.951i)4-s + (−0.457 + 0.189i)5-s + (−0.911 − 0.911i)7-s + (0.310 − 0.950i)8-s + (0.481 + 0.115i)10-s + (0.163 − 0.0676i)11-s + (1.28 + 0.532i)13-s + (0.201 + 1.27i)14-s + (−0.809 + 0.586i)16-s − 0.364i·17-s + (−0.447 + 1.08i)19-s + (−0.321 − 0.376i)20-s + (−0.171 − 0.0413i)22-s + (−1.02 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.196 - 0.980i$
Motivic weight: \(2\)
Character: $\chi_{288} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.196 - 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.384826 + 0.315197i\)
\(L(\frac12)\) \(\approx\) \(0.384826 + 0.315197i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 1.17i)T \)
3 \( 1 \)
good5 \( 1 + (2.28 - 0.948i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (6.37 + 6.37i)T + 49iT^{2} \)
11 \( 1 + (-1.79 + 0.744i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-16.7 - 6.91i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 6.19iT - 289T^{2} \)
19 \( 1 + (8.50 - 20.5i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (23.6 - 23.6i)T - 529iT^{2} \)
29 \( 1 + (14.5 - 35.1i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 14.1iT - 961T^{2} \)
37 \( 1 + (30.0 - 12.4i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-56.9 - 56.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-54.5 + 22.5i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 34.8T + 2.20e3T^{2} \)
53 \( 1 + (3.92 + 9.48i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (9.41 + 22.7i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-3.00 + 7.25i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (55.9 + 23.1i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (6.27 + 6.27i)T + 5.04e3iT^{2} \)
73 \( 1 + (-66.4 - 66.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.8T + 6.24e3T^{2} \)
83 \( 1 + (-1.23 + 2.97i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (36.7 - 36.7i)T - 7.92e3iT^{2} \)
97 \( 1 - 90.0T + 9.40e3T^{2} \)
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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

−11.53789337088879539779880813315, −10.83428723480698774356796371595, −9.949193832103369830920909090166, −9.113167361268035520315165795341, −8.005760042538368470845284626661, −7.14100569872787382619314486279, −6.15547556124430009254754700411, −3.95207653317501498865182342081, −3.42808209741735699036171313182, −1.44595610300953997345490891856, 0.33256655890586171559329125159, 2.39868164044455157498389677275, 4.14796513229400981048731144763, 5.85467920307685610963839318986, 6.29773762444625391268982937519, 7.67411929741712510538857051059, 8.597482835271052626757581940574, 9.223077847413388834140343411299, 10.30654872953376707593394362958, 11.20237602852972471261408672876

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