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Properties

Label 2-288-72.11-c1-0-0
Degree $2$
Conductor $288$
Sign $0.861 - 0.507i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 + 0.158i)3-s + (2.94 − 0.548i)9-s + (3.27 + 1.89i)11-s + 8.02i·17-s + 8.34·19-s + (2.5 − 4.33i)25-s + (−4.99 + 1.41i)27-s + (−5.94 − 2.74i)33-s + (−0.398 + 0.230i)41-s + (−1.17 + 2.03i)43-s + (−3.5 − 6.06i)49-s + (−1.27 − 13.8i)51-s + (−14.3 + 1.32i)57-s + (−10.6 + 6.13i)59-s + (−7.17 − 12.4i)67-s + ⋯
L(s)  = 1  + (−0.995 + 0.0917i)3-s + (0.983 − 0.182i)9-s + (0.987 + 0.570i)11-s + 1.94i·17-s + 1.91·19-s + (0.5 − 0.866i)25-s + (−0.962 + 0.272i)27-s + (−1.03 − 0.477i)33-s + (−0.0623 + 0.0359i)41-s + (−0.179 + 0.310i)43-s + (−0.5 − 0.866i)49-s + (−0.178 − 1.93i)51-s + (−1.90 + 0.175i)57-s + (−1.38 + 0.798i)59-s + (−0.876 − 1.51i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.861 - 0.507i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974615 + 0.265973i\)
\(L(\frac12)\) \(\approx\) \(0.974615 + 0.265973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.158i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.27 - 1.89i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 8.02iT - 17T^{2} \)
19 \( 1 - 8.34T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (0.398 - 0.230i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.17 - 2.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (10.6 - 6.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.17 + 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.44 - 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + (9.84 - 17.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.99313925735939708058797026571, −10.98145712057326684663790936462, −10.12143918116198690893488364072, −9.270312519896209347181193559372, −7.920531447084458532137297802157, −6.78457114150948124790182012393, −5.97829348138928179141918463939, −4.79838545457740215565684306812, −3.68849337934032410300149581862, −1.44872290482539340360860925926, 1.06162082961248363009660896400, 3.26201932961832851480031272467, 4.79068267244108707273614832376, 5.64910693417385723551842872037, 6.83099914402236818248649814665, 7.55329826700543950434768949724, 9.163979581773034340784629642008, 9.785038403739745332585007343305, 11.15509175516008389067823390551, 11.58975277535010450734568043171

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