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Properties

Label 2-288-72.11-c1-0-3
Degree $2$
Conductor $288$
Sign $0.973 + 0.228i$
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  + (−0.925 − 1.46i)3-s + (0.895 + 1.55i)5-s + (2.08 + 1.20i)7-s + (−1.28 + 2.71i)9-s + (1.36 + 0.790i)11-s + (5.35 − 3.09i)13-s + (1.44 − 2.74i)15-s − 3.69i·17-s − 3.12·19-s + (−0.167 − 4.17i)21-s + (1.36 + 2.35i)23-s + (0.896 − 1.55i)25-s + (5.15 − 0.625i)27-s + (−2.55 + 4.42i)29-s + (5.95 − 3.43i)31-s + ⋯
L(s)  = 1  + (−0.534 − 0.845i)3-s + (0.400 + 0.693i)5-s + (0.789 + 0.455i)7-s + (−0.428 + 0.903i)9-s + (0.412 + 0.238i)11-s + (1.48 − 0.857i)13-s + (0.372 − 0.709i)15-s − 0.897i·17-s − 0.717·19-s + (−0.0366 − 0.910i)21-s + (0.283 + 0.491i)23-s + (0.179 − 0.310i)25-s + (0.992 − 0.120i)27-s + (−0.474 + 0.821i)29-s + (1.06 − 0.617i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.973 + 0.228i$
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: \((2,\ 288,\ (\ :1/2),\ 0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28372 - 0.148894i\)
\(L(\frac12)\) \(\approx\) \(1.28372 - 0.148894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.925 + 1.46i)T \)
good5 \( 1 + (-0.895 - 1.55i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.08 - 1.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.36 - 0.790i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.35 + 3.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.69iT - 17T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
23 \( 1 + (-1.36 - 2.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.55 - 4.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.95 + 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.24iT - 37T^{2} \)
41 \( 1 + (5.32 - 3.07i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.452 + 0.783i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.88 - 8.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.05T + 53T^{2} \)
59 \( 1 + (-6.10 + 3.52i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.05 + 1.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.03 + 1.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.31T + 71T^{2} \)
73 \( 1 - 0.631T + 73T^{2} \)
79 \( 1 + (7.82 + 4.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.5 + 7.82i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.16iT - 89T^{2} \)
97 \( 1 + (6.72 - 11.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.55869376386877983842322137840, −11.12954649970630870581018523506, −10.12506500631819836715092676651, −8.690619859312973233082333952661, −7.88572475777551401325520418100, −6.69811938702294215245971768600, −5.98739295830040336002479836798, −4.86890986674098260417941993249, −2.95436559968820502108632481714, −1.50373819484859003868790503317, 1.41158412030786931712559856285, 3.81678025062914595216955401379, 4.58398655385416693741982236408, 5.76354804627576433605559129343, 6.64603788314061019699355108615, 8.447706290734207534574261336265, 8.879783468955236761534479848939, 10.08767124153486147288457562157, 10.97845322998147757311483258309, 11.55223879867174732996243864895

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