Properties

Label 2-288-72.67-c2-0-5
Degree $2$
Conductor $288$
Sign $-0.354 - 0.934i$
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  + (0.136 + 2.99i)3-s + (3.84 + 2.22i)5-s + (−0.704 + 0.406i)7-s + (−8.96 + 0.816i)9-s + (3.72 + 6.44i)11-s + (18.0 + 10.4i)13-s + (−6.13 + 11.8i)15-s + 1.74·17-s − 31.7·19-s + (−1.31 − 2.05i)21-s + (6.44 + 3.72i)23-s + (−2.64 − 4.57i)25-s + (−3.66 − 26.7i)27-s + (−26.9 + 15.5i)29-s + (4.91 + 2.83i)31-s + ⋯
L(s)  = 1  + (0.0453 + 0.998i)3-s + (0.769 + 0.444i)5-s + (−0.100 + 0.0580i)7-s + (−0.995 + 0.0906i)9-s + (0.338 + 0.585i)11-s + (1.39 + 0.803i)13-s + (−0.408 + 0.788i)15-s + 0.102·17-s − 1.67·19-s + (−0.0625 − 0.0978i)21-s + (0.280 + 0.161i)23-s + (−0.105 − 0.182i)25-s + (−0.135 − 0.990i)27-s + (−0.929 + 0.536i)29-s + (0.158 + 0.0915i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.354 - 0.934i$
Analytic conductor: \(7.84743\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.354 - 0.934i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.975816 + 1.41392i\)
\(L(\frac12)\) \(\approx\) \(0.975816 + 1.41392i\)
\(L(2)\) 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 \( 1 + (-0.136 - 2.99i)T \)
good5 \( 1 + (-3.84 - 2.22i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.704 - 0.406i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.72 - 6.44i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-18.0 - 10.4i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 1.74T + 289T^{2} \)
19 \( 1 + 31.7T + 361T^{2} \)
23 \( 1 + (-6.44 - 3.72i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (26.9 - 15.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-4.91 - 2.83i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 62.0iT - 1.36e3T^{2} \)
41 \( 1 + (-2.74 + 4.74i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.3 - 38.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-71.1 + 41.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 85.0iT - 2.80e3T^{2} \)
59 \( 1 + (-21.8 + 37.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-61.1 + 35.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.91 - 17.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 69.3iT - 5.04e3T^{2} \)
73 \( 1 + 21.7T + 5.32e3T^{2} \)
79 \( 1 + (-37.1 + 21.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-3.53 - 6.11i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 37.1T + 7.92e3T^{2} \)
97 \( 1 + (-9.38 - 16.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.57364811958135234757369297587, −10.81594675861057256871870980438, −10.02011530071869390099890589279, −9.165012564427605926810768931379, −8.374588116235198753812509442225, −6.66464040262407688044243086698, −5.97556028884495290841397051395, −4.60823233887833574063557590973, −3.57432472030320284122060450355, −2.04451115318233243399206193831, 0.875308445231434575782919139042, 2.24089351360293712514604819002, 3.81515154533358876697294824193, 5.73482020314994967200211257389, 6.08025197081067081158545299337, 7.39651048609358518287874862368, 8.558280024399423780229601155413, 9.050835854668750002608061641943, 10.55903085502372925969133535584, 11.26970368165209230340297920203

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