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Properties

Label 2-288-32.21-c1-0-7
Degree $2$
Conductor $288$
Sign $0.679 + 0.733i$
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  + (−0.582 − 1.28i)2-s + (−1.32 + 1.50i)4-s + (0.445 − 1.07i)5-s + (2.57 + 2.57i)7-s + (2.70 + 0.829i)8-s + (−1.64 + 0.0521i)10-s + (2.20 + 0.914i)11-s + (0.264 + 0.638i)13-s + (1.81 − 4.81i)14-s + (−0.505 − 3.96i)16-s − 2.58i·17-s + (−0.695 − 1.67i)19-s + (1.02 + 2.09i)20-s + (−0.106 − 3.37i)22-s + (5.69 − 5.69i)23-s + ⋯
L(s)  = 1  + (−0.411 − 0.911i)2-s + (−0.660 + 0.750i)4-s + (0.199 − 0.481i)5-s + (0.973 + 0.973i)7-s + (0.956 + 0.293i)8-s + (−0.520 + 0.0164i)10-s + (0.665 + 0.275i)11-s + (0.0733 + 0.177i)13-s + (0.486 − 1.28i)14-s + (−0.126 − 0.991i)16-s − 0.626i·17-s + (−0.159 − 0.385i)19-s + (0.229 + 0.467i)20-s + (−0.0228 − 0.719i)22-s + (1.18 − 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.679 + 0.733i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06421 - 0.464702i\)
\(L(\frac12)\) \(\approx\) \(1.06421 - 0.464702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.582 + 1.28i)T \)
3 \( 1 \)
good5 \( 1 + (-0.445 + 1.07i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.57 - 2.57i)T + 7iT^{2} \)
11 \( 1 + (-2.20 - 0.914i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.264 - 0.638i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.58iT - 17T^{2} \)
19 \( 1 + (0.695 + 1.67i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.69 + 5.69i)T - 23iT^{2} \)
29 \( 1 + (7.43 - 3.07i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.28T + 31T^{2} \)
37 \( 1 + (-2.64 + 6.37i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.15 - 7.15i)T - 41iT^{2} \)
43 \( 1 + (1.64 + 0.681i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.69iT - 47T^{2} \)
53 \( 1 + (5.90 + 2.44i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.43 + 3.47i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-8.51 + 3.52i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (8.92 - 3.69i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-2.02 - 2.02i)T + 71iT^{2} \)
73 \( 1 + (10.7 - 10.7i)T - 73iT^{2} \)
79 \( 1 + 0.523iT - 79T^{2} \)
83 \( 1 + (-4.73 - 11.4i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.44 + 9.44i)T + 89iT^{2} \)
97 \( 1 + 4.39T + 97T^{2} \)
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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

−11.51550783643982779410947607621, −11.04267380161531086993254336298, −9.639199161516941097472737315261, −8.948454308236101391007039996244, −8.295026647378379333285984600845, −6.96690138517344066926570479310, −5.26195150592556957754143800733, −4.46955627338884247183182932922, −2.76925679205951854318873316665, −1.45364260943029393319959163711, 1.38611047343044395910724843359, 3.80097973848297558496086818334, 4.97638981405574994069666900167, 6.18657026024774457737304893980, 7.13011357737024960553156308622, 7.963899637057296735389407002301, 8.894867677419621049688810553448, 10.07391301328346728101868776225, 10.75381392338101093083678809099, 11.67735104378446243519873726543

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