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Properties

Label 2-288-288.11-c1-0-17
Degree $2$
Conductor $288$
Sign $0.965 + 0.260i$
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  + (−0.320 − 1.37i)2-s + (1.54 − 0.775i)3-s + (−1.79 + 0.882i)4-s + (0.473 + 3.59i)5-s + (−1.56 − 1.88i)6-s + (0.771 + 2.87i)7-s + (1.79 + 2.18i)8-s + (1.79 − 2.40i)9-s + (4.80 − 1.80i)10-s + (0.294 − 0.225i)11-s + (−2.09 + 2.75i)12-s + (−1.23 + 1.60i)13-s + (3.71 − 1.98i)14-s + (3.52 + 5.20i)15-s + (2.44 − 3.16i)16-s − 1.73·17-s + ⋯
L(s)  = 1  + (−0.226 − 0.973i)2-s + (0.894 − 0.447i)3-s + (−0.897 + 0.441i)4-s + (0.211 + 1.60i)5-s + (−0.638 − 0.769i)6-s + (0.291 + 1.08i)7-s + (0.633 + 0.774i)8-s + (0.599 − 0.800i)9-s + (1.51 − 0.570i)10-s + (0.0888 − 0.0681i)11-s + (−0.604 + 0.796i)12-s + (−0.342 + 0.446i)13-s + (0.993 − 0.530i)14-s + (0.909 + 1.34i)15-s + (0.610 − 0.792i)16-s − 0.420·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.965 + 0.260i$
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: \((2,\ 288,\ (\ :1/2),\ 0.965 + 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45627 - 0.193035i\)
\(L(\frac12)\) \(\approx\) \(1.45627 - 0.193035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.320 + 1.37i)T \)
3 \( 1 + (-1.54 + 0.775i)T \)
good5 \( 1 + (-0.473 - 3.59i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.771 - 2.87i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.294 + 0.225i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (1.23 - 1.60i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + (-6.43 + 2.66i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (6.84 + 1.83i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.47 + 0.852i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (-4.77 + 2.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.42 + 5.85i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.78 + 6.67i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.20 - 5.47i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (-8.17 - 4.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.978 + 2.36i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.17 - 0.681i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (-1.10 + 8.40i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-2.35 + 3.06i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-0.815 - 0.815i)T + 71iT^{2} \)
73 \( 1 + (7.29 - 7.29i)T - 73iT^{2} \)
79 \( 1 + (3.00 - 5.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.71 - 1.27i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (8.94 - 8.94i)T - 89iT^{2} \)
97 \( 1 + (3.73 - 6.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.70994395614102262966067688848, −10.94826348374186857467037976080, −9.721810772634246194867423681300, −9.235868639931728068552020180246, −8.005348238709622861301802403899, −7.17305119053032363777549162386, −5.83964100974840730883339580239, −3.98612269930697506695441584495, −2.73547995150583102078680585995, −2.17630256453809579656167897954, 1.29199745025964402123505036130, 3.91040898040224661376196349231, 4.70466204542366171699246846507, 5.66211084861726556361044919810, 7.45000595890959218537314485862, 7.957551342596535801767891414970, 8.904795023567692686155256409896, 9.690314005372248591295913201907, 10.34875571869378343342683128081, 12.08611853448151766310974532858

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