Properties

Label 2-288-32.13-c1-0-3
Degree $2$
Conductor $288$
Sign $-0.812 - 0.583i$
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  + (1.06 + 0.925i)2-s + (0.286 + 1.97i)4-s + (−3.41 + 1.41i)5-s + (−1.42 + 1.42i)7-s + (−1.52 + 2.38i)8-s + (−4.95 − 1.64i)10-s + (−0.821 − 1.98i)11-s + (3.77 + 1.56i)13-s + (−2.85 + 0.205i)14-s + (−3.83 + 1.13i)16-s + 0.438i·17-s + (2.08 + 0.865i)19-s + (−3.77 − 6.35i)20-s + (0.957 − 2.88i)22-s + (4.42 + 4.42i)23-s + ⋯
L(s)  = 1  + (0.756 + 0.654i)2-s + (0.143 + 0.989i)4-s + (−1.52 + 0.632i)5-s + (−0.540 + 0.540i)7-s + (−0.539 + 0.841i)8-s + (−1.56 − 0.521i)10-s + (−0.247 − 0.598i)11-s + (1.04 + 0.433i)13-s + (−0.762 + 0.0548i)14-s + (−0.958 + 0.283i)16-s + 0.106i·17-s + (0.479 + 0.198i)19-s + (−0.844 − 1.42i)20-s + (0.204 − 0.614i)22-s + (0.922 + 0.922i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.812 - 0.583i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387780 + 1.20534i\)
\(L(\frac12)\) \(\approx\) \(0.387780 + 1.20534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 - 0.925i)T \)
3 \( 1 \)
good5 \( 1 + (3.41 - 1.41i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.42 - 1.42i)T - 7iT^{2} \)
11 \( 1 + (0.821 + 1.98i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-3.77 - 1.56i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 0.438iT - 17T^{2} \)
19 \( 1 + (-2.08 - 0.865i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.42 - 4.42i)T + 23iT^{2} \)
29 \( 1 + (2.48 - 6.00i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + (8.11 - 3.36i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.71 - 3.71i)T + 41iT^{2} \)
43 \( 1 + (4.47 + 10.8i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.94iT - 47T^{2} \)
53 \( 1 + (1.97 + 4.77i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.54 + 1.05i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.65 - 6.42i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.876 - 2.11i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-8.63 + 8.63i)T - 71iT^{2} \)
73 \( 1 + (3.27 + 3.27i)T + 73iT^{2} \)
79 \( 1 - 2.97iT - 79T^{2} \)
83 \( 1 + (-6.25 - 2.58i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.0975 - 0.0975i)T - 89iT^{2} \)
97 \( 1 - 7.26T + 97T^{2} \)
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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

−12.04441721237500430655241106934, −11.59724699442217593485418985319, −10.66471559637235455641818903238, −8.910849226674188360788121750413, −8.193146165019643497903308327480, −7.16861516347981947122222088907, −6.38109753647315336948515561566, −5.14047854394971612266759638992, −3.69274420570450023711332942727, −3.15346390004814906958054198623, 0.78054632955560180640889973827, 3.08377094804346124019452964603, 4.09200013533244493785546807255, 4.88792647337683891458768817576, 6.38012027339491257393169969249, 7.50237469647230118839709883218, 8.579732006148008529498854058954, 9.774643089333532507415912047629, 10.80544228334514786485164956359, 11.53231197319036249099462780028

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