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Properties

Label 2-288-72.11-c1-0-9
Degree $2$
Conductor $288$
Sign $-0.999 + 0.0426i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.999 + 0.0426i$
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: \((2,\ 288,\ (\ :1/2),\ -0.999 + 0.0426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00935729 - 0.438575i\)
\(L(\frac12)\) \(\approx\) \(0.00935729 - 0.438575i\)
\(L(\frac{3}{2})\) 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

−11.74686109844100677768064665183, −10.44136363840464243525744220220, −9.455807260653639393150244110449, −8.382106118406701657980172388598, −7.15469899351030587221842615278, −6.70204280773861039337114296089, −5.22919076229019038382395861499, −4.20568993597363590955386023906, −2.28371379343015997617721740178, −0.33117559347902012080451403927, 2.88943942536569510494639178436, 3.89765445901448578196373356571, 5.28676199481785924811700315030, 6.25296676721690303966125924115, 7.27025509851689144859895088309, 8.644349316046093625362871670805, 9.680096633070201922203601825311, 10.38582484026199025674287841852, 11.21278767180066038271993054902, 12.21655517064921922214033044683

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