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Properties

Label 2-288-72.59-c1-0-2
Degree $2$
Conductor $288$
Sign $-0.248 - 0.968i$
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  + (1.12 + 1.31i)3-s + (−0.565 + 0.978i)5-s + (−3.71 + 2.14i)7-s + (−0.456 + 2.96i)9-s + (−1.00 + 0.582i)11-s + (2.64 + 1.52i)13-s + (−1.92 + 0.360i)15-s − 1.49i·17-s + 3.42·19-s + (−7.00 − 2.46i)21-s + (3.85 − 6.68i)23-s + (1.86 + 3.22i)25-s + (−4.41 + 2.74i)27-s + (−0.709 − 1.22i)29-s + (4.66 + 2.69i)31-s + ⋯
L(s)  = 1  + (0.651 + 0.758i)3-s + (−0.252 + 0.437i)5-s + (−1.40 + 0.810i)7-s + (−0.152 + 0.988i)9-s + (−0.304 + 0.175i)11-s + (0.733 + 0.423i)13-s + (−0.496 + 0.0932i)15-s − 0.362i·17-s + 0.785·19-s + (−1.52 − 0.537i)21-s + (0.804 − 1.39i)23-s + (0.372 + 0.644i)25-s + (−0.849 + 0.528i)27-s + (−0.131 − 0.228i)29-s + (0.837 + 0.483i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.248 - 0.968i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771588 + 0.995053i\)
\(L(\frac12)\) \(\approx\) \(0.771588 + 0.995053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 1.31i)T \)
good5 \( 1 + (0.565 - 0.978i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.71 - 2.14i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 - 0.582i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.64 - 1.52i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.49iT - 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 + (-3.85 + 6.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.709 + 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.66 - 2.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.97iT - 37T^{2} \)
41 \( 1 + (4.23 + 2.44i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.74 - 3.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.77 - 3.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (7.50 + 4.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.58 + 9.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.54T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 + (-2.24 + 1.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.98 + 2.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.63iT - 89T^{2} \)
97 \( 1 + (-3.35 - 5.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.07694097594537687652517039318, −10.97040009633041428936891668177, −10.08839594305945134914006260717, −9.239387516516301910287109236033, −8.564704575417333367736959616250, −7.21230363650804228842884378458, −6.16912098199995709740061897294, −4.85685223422974779647419046809, −3.42779416828298978990196185443, −2.70497940532863656314890070475, 0.922414244749295670789846751327, 3.01346848289248341381324917959, 3.86967709464924223776589892710, 5.71666913866803798245570152656, 6.79438244916040441467856326656, 7.61261684355275686344555668011, 8.630256482537618623289410308087, 9.532939501809812866287906321859, 10.47929896458922371017863205442, 11.75927001517593996157502478670

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