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Properties

Label 2-288-96.11-c1-0-10
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: $\chi_{288} (107, \cdot )$
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

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

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