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Properties

Label 2-288-9.4-c3-0-6
Degree $2$
Conductor $288$
Sign $-0.880 - 0.474i$
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  + (−0.378 + 5.18i)3-s + (2.25 − 3.90i)5-s + (10.1 + 17.5i)7-s + (−26.7 − 3.91i)9-s + (4.74 + 8.21i)11-s + (−18.5 + 32.0i)13-s + (19.4 + 13.1i)15-s − 11.9·17-s + 24.3·19-s + (−94.9 + 45.9i)21-s + (−10.3 + 17.9i)23-s + (52.3 + 90.6i)25-s + (30.4 − 136. i)27-s + (−36.8 − 63.8i)29-s + (−112. + 195. i)31-s + ⋯
L(s)  = 1  + (−0.0727 + 0.997i)3-s + (0.201 − 0.349i)5-s + (0.548 + 0.949i)7-s + (−0.989 − 0.145i)9-s + (0.129 + 0.225i)11-s + (−0.394 + 0.683i)13-s + (0.334 + 0.226i)15-s − 0.169·17-s + 0.293·19-s + (−0.987 + 0.477i)21-s + (−0.0938 + 0.162i)23-s + (0.418 + 0.724i)25-s + (0.216 − 0.976i)27-s + (−0.236 − 0.409i)29-s + (−0.654 + 1.13i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.880 - 0.474i$
Analytic conductor: \(16.9925\)
Root analytic conductor: \(4.12220\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :3/2),\ -0.880 - 0.474i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.316647334\)
\(L(\frac12)\) \(\approx\) \(1.316647334\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.378 - 5.18i)T \)
good5 \( 1 + (-2.25 + 3.90i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-10.1 - 17.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-4.74 - 8.21i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (18.5 - 32.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 11.9T + 4.91e3T^{2} \)
19 \( 1 - 24.3T + 6.85e3T^{2} \)
23 \( 1 + (10.3 - 17.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (36.8 + 63.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (112. - 195. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 377.T + 5.06e4T^{2} \)
41 \( 1 + (195. - 339. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (110. + 191. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (24.4 + 42.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 433.T + 1.48e5T^{2} \)
59 \( 1 + (-166. + 288. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (191. + 331. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (374. - 648. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 897.T + 3.57e5T^{2} \)
73 \( 1 - 105.T + 3.89e5T^{2} \)
79 \( 1 + (-229. - 397. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-729. - 1.26e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 979.T + 7.04e5T^{2} \)
97 \( 1 + (-476. - 825. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.70957986412929941359747033294, −10.86593270835864175672941778823, −9.719990035408130910879570321191, −9.059263965415670431488803874033, −8.269945169936525319381010745428, −6.75454203253847766986971359369, −5.38030966092289141848162537230, −4.86037670345941022876849875455, −3.44042064750902486051113350906, −1.91507462304289297653160344466, 0.48465659471948971209837239183, 1.91185004860752768306850662283, 3.35489727785398270899451974242, 4.95505456340288316848992159477, 6.14569501738867692411352201059, 7.17159657772793008116681841271, 7.82178509235679802164730537023, 8.887004705752825633347974221146, 10.30446671130204958244889286017, 10.96999582960669448842875808330

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