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Properties

Label 2-288-32.5-c1-0-14
Degree $2$
Conductor $288$
Sign $0.637 + 0.770i$
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  + (1.29 − 0.567i)2-s + (1.35 − 1.47i)4-s + (−0.825 − 0.342i)5-s + (1.17 + 1.17i)7-s + (0.920 − 2.67i)8-s + (−1.26 + 0.0257i)10-s + (1.46 − 3.53i)11-s + (3.01 − 1.24i)13-s + (2.19 + 0.856i)14-s + (−0.325 − 3.98i)16-s + 4.58i·17-s + (−3.29 + 1.36i)19-s + (−1.62 + 0.750i)20-s + (−0.110 − 5.41i)22-s + (−5.41 + 5.41i)23-s + ⋯
L(s)  = 1  + (0.915 − 0.401i)2-s + (0.677 − 0.735i)4-s + (−0.369 − 0.152i)5-s + (0.445 + 0.445i)7-s + (0.325 − 0.945i)8-s + (−0.399 + 0.00814i)10-s + (0.441 − 1.06i)11-s + (0.834 − 0.345i)13-s + (0.586 + 0.228i)14-s + (−0.0814 − 0.996i)16-s + 1.11i·17-s + (−0.757 + 0.313i)19-s + (−0.362 + 0.167i)20-s + (−0.0235 − 1.15i)22-s + (−1.12 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93825 - 0.912327i\)
\(L(\frac12)\) \(\approx\) \(1.93825 - 0.912327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 0.567i)T \)
3 \( 1 \)
good5 \( 1 + (0.825 + 0.342i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.17 - 1.17i)T + 7iT^{2} \)
11 \( 1 + (-1.46 + 3.53i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-3.01 + 1.24i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 4.58iT - 17T^{2} \)
19 \( 1 + (3.29 - 1.36i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.41 - 5.41i)T - 23iT^{2} \)
29 \( 1 + (-2.46 - 5.95i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 + (-7.33 - 3.03i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.35 + 1.35i)T - 41iT^{2} \)
43 \( 1 + (2.95 - 7.14i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.16iT - 47T^{2} \)
53 \( 1 + (-3.13 + 7.57i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.221 + 0.0919i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.66 - 6.44i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.52 + 13.3i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.51 + 1.51i)T + 71iT^{2} \)
73 \( 1 + (9.62 - 9.62i)T - 73iT^{2} \)
79 \( 1 + 5.34iT - 79T^{2} \)
83 \( 1 + (5.64 - 2.33i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.09 + 5.09i)T + 89iT^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−11.64256457899711403515986828483, −11.08882505535504681602653186170, −10.11150519071794489094873473081, −8.724220267516289390505458097261, −7.891827648627466396357224208586, −6.26026596947120411820007079486, −5.71416174514331921963766713635, −4.25171441319893286042687297616, −3.36327156883812412574500959991, −1.61952733373504488010850198995, 2.21692088587788632902439381442, 3.94715713006003703508792296681, 4.55581810270659257023736352311, 5.99356165484438865545172050638, 7.01738223574601056518379167730, 7.75665325927952181086713600190, 8.896438994675106615543532872488, 10.27959155819420556006647669417, 11.36283756097881551108512312507, 11.91611472473129077086321999332

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