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Properties

Label 2-288-288.11-c1-0-11
Degree $2$
Conductor $288$
Sign $0.748 + 0.662i$
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  + (−0.200 − 1.39i)2-s + (−1.56 + 0.751i)3-s + (−1.91 + 0.562i)4-s + (−0.202 − 1.54i)5-s + (1.36 + 2.03i)6-s + (1.17 + 4.39i)7-s + (1.17 + 2.57i)8-s + (1.87 − 2.34i)9-s + (−2.11 + 0.593i)10-s + (3.93 − 3.02i)11-s + (2.57 − 2.31i)12-s + (0.682 − 0.888i)13-s + (5.91 − 2.53i)14-s + (1.47 + 2.25i)15-s + (3.36 − 2.15i)16-s − 4.08·17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (−0.901 + 0.433i)3-s + (−0.959 + 0.281i)4-s + (−0.0906 − 0.688i)5-s + (0.557 + 0.830i)6-s + (0.445 + 1.66i)7-s + (0.414 + 0.910i)8-s + (0.623 − 0.781i)9-s + (−0.668 + 0.187i)10-s + (1.18 − 0.910i)11-s + (0.742 − 0.669i)12-s + (0.189 − 0.246i)13-s + (1.58 − 0.676i)14-s + (0.380 + 0.581i)15-s + (0.841 − 0.539i)16-s − 0.990·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.748 + 0.662i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.748 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846853 - 0.320927i\)
\(L(\frac12)\) \(\approx\) \(0.846853 - 0.320927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.200 + 1.39i)T \)
3 \( 1 + (1.56 - 0.751i)T \)
good5 \( 1 + (0.202 + 1.54i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-1.17 - 4.39i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.93 + 3.02i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-0.682 + 0.888i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 4.08T + 17T^{2} \)
19 \( 1 + (-2.37 + 0.985i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.35 - 1.43i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-9.39 - 1.23i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (0.512 - 0.296i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.95 - 4.70i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.25 - 4.67i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.63 - 2.12i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (2.20 + 1.27i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.74 + 6.61i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.630 - 0.0830i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (-1.61 + 12.2i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (5.85 - 7.62i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-1.64 - 1.64i)T + 71iT^{2} \)
73 \( 1 + (0.361 - 0.361i)T - 73iT^{2} \)
79 \( 1 + (3.24 - 5.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.0 + 1.58i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (2.45 - 2.45i)T - 89iT^{2} \)
97 \( 1 + (-8.51 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.52276974999297731254430991928, −11.21904259051978850522461957734, −9.841574296548295988687650707953, −8.770584357553734719111211774795, −8.663269214940212151538499324193, −6.45707050354681358937504889167, −5.29799207477466056491633794912, −4.61674709793666561415517373350, −3.09418131497820130639809864448, −1.19757068644463131135446496771, 1.14208155889266659542537721082, 4.08957245366209121769255845369, 4.80906837507603011307466351377, 6.42974199887864150517926411499, 7.00155589524207998211098680901, 7.48287736190258522457146179367, 8.947458623607146204364109852121, 10.26169588617088542566013122337, 10.77080276199235000085108418910, 11.86782643775720054270714968914

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