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Properties

Label 2-288-96.35-c1-0-5
Degree $2$
Conductor $288$
Sign $-0.0820 - 0.996i$
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  + (0.345 + 1.37i)2-s + (−1.76 + 0.947i)4-s + (1.64 − 0.682i)5-s + (2.51 + 2.51i)7-s + (−1.90 − 2.08i)8-s + (1.50 + 2.02i)10-s + (2.69 − 1.11i)11-s + (−1.76 + 4.25i)13-s + (−2.57 + 4.31i)14-s + (2.20 − 3.33i)16-s − 6.10·17-s + (3.43 + 1.42i)19-s + (−2.25 + 2.76i)20-s + (2.45 + 3.30i)22-s + (0.525 + 0.525i)23-s + ⋯
L(s)  = 1  + (0.244 + 0.969i)2-s + (−0.880 + 0.473i)4-s + (0.737 − 0.305i)5-s + (0.949 + 0.949i)7-s + (−0.674 − 0.738i)8-s + (0.476 + 0.640i)10-s + (0.811 − 0.336i)11-s + (−0.488 + 1.17i)13-s + (−0.688 + 1.15i)14-s + (0.551 − 0.834i)16-s − 1.47·17-s + (0.787 + 0.326i)19-s + (−0.504 + 0.618i)20-s + (0.524 + 0.704i)22-s + (0.109 + 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.0820 - 0.996i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.0820 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06237 + 1.15344i\)
\(L(\frac12)\) \(\approx\) \(1.06237 + 1.15344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.345 - 1.37i)T \)
3 \( 1 \)
good5 \( 1 + (-1.64 + 0.682i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.51 - 2.51i)T + 7iT^{2} \)
11 \( 1 + (-2.69 + 1.11i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.76 - 4.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.10T + 17T^{2} \)
19 \( 1 + (-3.43 - 1.42i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.525 - 0.525i)T + 23iT^{2} \)
29 \( 1 + (-1.46 + 3.53i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.55iT - 31T^{2} \)
37 \( 1 + (-2.30 - 5.56i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.04 + 3.04i)T - 41iT^{2} \)
43 \( 1 + (3.31 + 8.00i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 8.59iT - 47T^{2} \)
53 \( 1 + (3.78 + 9.14i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.73 - 9.01i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.41 - 1.41i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-0.0538 + 0.130i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.31 + 1.31i)T - 71iT^{2} \)
73 \( 1 + (10.9 + 10.9i)T + 73iT^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (4.38 - 10.5i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.97 + 5.97i)T + 89iT^{2} \)
97 \( 1 + 3.21T + 97T^{2} \)
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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

−11.95203339465387768788089447824, −11.52707258648115814421840465183, −9.696272296383797664112059814774, −9.053714978851238015081739734236, −8.324366681365683435645619827614, −7.02435122929998933288067119357, −6.04881303363934558844944392769, −5.15406526756712552015410709434, −4.13746423592392871392600028226, −2.06525774017315150017306409420, 1.34239853056112050675333635004, 2.77466804424986428611114589791, 4.27777471153072491943393418280, 5.14353119012086451119684022447, 6.51945585028744709517657152010, 7.78724752166942693526647734566, 9.020216769127111937285083264693, 9.923522320624281176653073388268, 10.76055932739944975526940838445, 11.33489393717435939030820008155

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