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Properties

Label 2-288-72.61-c1-0-2
Degree $2$
Conductor $288$
Sign $-0.932 - 0.360i$
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  + (0.294 + 1.70i)3-s + (−3.17 + 1.83i)5-s + (0.191 − 0.332i)7-s + (−2.82 + 1.00i)9-s + (−1.73 − 1.00i)11-s + (0.397 − 0.229i)13-s + (−4.06 − 4.87i)15-s − 4.08·17-s + 4.72i·19-s + (0.623 + 0.229i)21-s + (2.97 + 5.15i)23-s + (4.21 − 7.29i)25-s + (−2.54 − 4.52i)27-s + (2.03 + 1.17i)29-s + (−0.592 − 1.02i)31-s + ⋯
L(s)  = 1  + (0.170 + 0.985i)3-s + (−1.41 + 0.819i)5-s + (0.0725 − 0.125i)7-s + (−0.942 + 0.335i)9-s + (−0.524 − 0.302i)11-s + (0.110 − 0.0636i)13-s + (−1.04 − 1.25i)15-s − 0.990·17-s + 1.08i·19-s + (0.136 + 0.0501i)21-s + (0.620 + 1.07i)23-s + (0.842 − 1.45i)25-s + (−0.490 − 0.871i)27-s + (0.378 + 0.218i)29-s + (−0.106 − 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128448 + 0.688836i\)
\(L(\frac12)\) \(\approx\) \(0.128448 + 0.688836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.294 - 1.70i)T \)
good5 \( 1 + (3.17 - 1.83i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.191 + 0.332i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 + 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.397 + 0.229i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.08T + 17T^{2} \)
19 \( 1 - 4.72iT - 19T^{2} \)
23 \( 1 + (-2.97 - 5.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 1.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.592 + 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.74iT - 37T^{2} \)
41 \( 1 + (-4.75 - 8.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.03 - 0.598i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.27 - 5.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.63iT - 53T^{2} \)
59 \( 1 + (-0.603 + 0.348i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.23 + 2.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.87 + 5.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + (-5.35 + 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.49 + 3.16i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.56T + 89T^{2} \)
97 \( 1 + (2.98 - 5.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.84047953424509185642311620985, −11.13672403226187428730824093792, −10.59203421526250780453433459173, −9.470360209595932015819349874998, −8.266520913334500360253556261845, −7.65658517632643316460439583623, −6.31784270782429189328473237848, −4.86426700416628999617085539841, −3.82075537188615018485215052372, −2.95966657847869090049244502061, 0.50486468141228364867043829954, 2.50731636892248315609133652533, 4.10922635170233914066881551521, 5.20845130644421672802513887090, 6.77847968364019160879771676664, 7.51712891796636429874782539957, 8.527165958507300521928495068251, 8.981414248505543946575860380990, 10.82370556725406581893598390322, 11.56269101476698357910205681066

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