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Properties

Label 2-288-9.7-c3-0-16
Degree $2$
Conductor $288$
Sign $0.455 - 0.890i$
Analytic cond. $16.9925$
Root an. cond. $4.12220$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.83 − 1.90i)3-s + (7.51 + 13.0i)5-s + (−4.38 + 7.58i)7-s + (19.7 − 18.3i)9-s + (−18.5 + 32.1i)11-s + (18.7 + 32.5i)13-s + (61.0 + 48.6i)15-s + 16.9·17-s − 79.0·19-s + (−6.76 + 45.0i)21-s + (57.7 + 99.9i)23-s + (−50.4 + 87.3i)25-s + (60.6 − 126. i)27-s + (−90.0 + 155. i)29-s + (−108. − 187. i)31-s + ⋯
L(s)  = 1  + (0.930 − 0.365i)3-s + (0.672 + 1.16i)5-s + (−0.236 + 0.409i)7-s + (0.732 − 0.680i)9-s + (−0.508 + 0.880i)11-s + (0.400 + 0.694i)13-s + (1.05 + 0.837i)15-s + 0.242·17-s − 0.953·19-s + (−0.0702 + 0.467i)21-s + (0.523 + 0.906i)23-s + (−0.403 + 0.698i)25-s + (0.432 − 0.901i)27-s + (−0.576 + 0.998i)29-s + (−0.628 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(16.9925\)
Root analytic conductor: \(4.12220\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :3/2),\ 0.455 - 0.890i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.638450182\)
\(L(\frac12)\) \(\approx\) \(2.638450182\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.83 + 1.90i)T \)
good5 \( 1 + (-7.51 - 13.0i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (4.38 - 7.58i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (18.5 - 32.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-18.7 - 32.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 16.9T + 4.91e3T^{2} \)
19 \( 1 + 79.0T + 6.85e3T^{2} \)
23 \( 1 + (-57.7 - 99.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (90.0 - 155. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (108. + 187. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 265.T + 5.06e4T^{2} \)
41 \( 1 + (-23.2 - 40.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-0.305 + 0.529i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-244. + 422. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 754.T + 1.48e5T^{2} \)
59 \( 1 + (302. + 524. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (49.5 - 85.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-506. - 878. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 317.T + 3.57e5T^{2} \)
73 \( 1 + 905.T + 3.89e5T^{2} \)
79 \( 1 + (345. - 598. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-564. + 977. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 329.T + 7.04e5T^{2} \)
97 \( 1 + (-606. + 1.04e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.49604355326999096947709253346, −10.41456305610729009000866623021, −9.610986585899269740455621791045, −8.801245542599034482861972342126, −7.47461277716566550523044162289, −6.83306850323557612340685604733, −5.75850821310465432759821312160, −3.99855763934346591425271267326, −2.72373233522131761151654276137, −1.89566343049611467087210450025, 0.912597807435421485415321261566, 2.51026503951263402198662484091, 3.85490820244163059657654483611, 5.00351152133954052299173738468, 6.06134271990552198619464885377, 7.63307547922662896569048581537, 8.570090828290786525486744910421, 9.102572801754248452942877736767, 10.21201008850950950960356242294, 10.86801982963687498392058346082

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