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Properties

Label 2-288-9.5-c2-0-18
Degree $2$
Conductor $288$
Sign $0.0541 + 0.998i$
Analytic cond. $7.84743$
Root an. cond. $2.80132$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.65 − 2.50i)3-s + (−0.721 − 0.416i)5-s + (1.26 + 2.18i)7-s + (−3.53 − 8.27i)9-s + (9.47 − 5.47i)11-s + (4.36 − 7.55i)13-s + (−2.23 + 1.11i)15-s − 20.8i·17-s + 1.50·19-s + (7.54 + 0.454i)21-s + (−1.00 − 0.578i)23-s + (−12.1 − 21.0i)25-s + (−26.5 − 4.84i)27-s + (15.7 − 9.08i)29-s + (−25.6 + 44.4i)31-s + ⋯
L(s)  = 1  + (0.551 − 0.834i)3-s + (−0.144 − 0.0832i)5-s + (0.180 + 0.311i)7-s + (−0.392 − 0.919i)9-s + (0.861 − 0.497i)11-s + (0.335 − 0.581i)13-s + (−0.148 + 0.0744i)15-s − 1.22i·17-s + 0.0790·19-s + (0.359 + 0.0216i)21-s + (−0.0435 − 0.0251i)23-s + (−0.486 − 0.842i)25-s + (−0.983 − 0.179i)27-s + (0.542 − 0.313i)29-s + (−0.828 + 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0541 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0541 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.0541 + 0.998i$
Analytic conductor: \(7.84743\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.0541 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36405 - 1.29208i\)
\(L(\frac12)\) \(\approx\) \(1.36405 - 1.29208i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.65 + 2.50i)T \)
good5 \( 1 + (0.721 + 0.416i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.26 - 2.18i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-9.47 + 5.47i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.36 + 7.55i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 20.8iT - 289T^{2} \)
19 \( 1 - 1.50T + 361T^{2} \)
23 \( 1 + (1.00 + 0.578i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.7 + 9.08i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (25.6 - 44.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 7.93T + 1.36e3T^{2} \)
41 \( 1 + (21.8 + 12.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-19.3 - 33.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-59.6 + 34.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 46.5iT - 2.80e3T^{2} \)
59 \( 1 + (-89.1 - 51.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-44.1 - 76.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (11.3 - 19.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 + 75.2T + 5.32e3T^{2} \)
79 \( 1 + (-51.8 - 89.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-53.7 + 31.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 1.95iT - 7.92e3T^{2} \)
97 \( 1 + (-59.2 - 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.77041594320490341894844362128, −10.43569762235900042697370618214, −9.119944672201056336126347247690, −8.556108345297308538265215942882, −7.49292662999408637724223377231, −6.55683502246017100948248609710, −5.44301010326337264252055242743, −3.79621811040214847372365891871, −2.56725066009138782570532356211, −0.941710510117626153825832322200, 1.89323653071420306206757621063, 3.66133807437994801453107610582, 4.30188262827769741220181762365, 5.71334670687744038958645764947, 7.03957864242237893387340549505, 8.128764701316907144991272817042, 9.065641774140235036699802400404, 9.846287664789120169210582511175, 10.84328637659566306908796197078, 11.58187073289770021311665455327

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