Properties

Label 2-288-36.7-c0-0-0
Degree $2$
Conductor $288$
Sign $0.422 - 0.906i$
Analytic cond. $0.143730$
Root an. cond. $0.379118$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s − 9-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s − 0.999i·35-s + (0.866 − 0.5i)39-s + ⋯
L(s)  = 1  + i·3-s + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s − 9-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s − 0.999i·35-s + (0.866 − 0.5i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.422 - 0.906i$
Analytic conductor: \(0.143730\)
Root analytic conductor: \(0.379118\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :0),\ 0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7632674813\)
\(L(\frac12)\) \(\approx\) \(0.7632674813\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−12.11185467201899770538948583746, −10.96788935674054466893427629186, −10.18249909288733876997531176562, −9.735895127543023512766095797925, −8.649405902211235628701397942728, −7.12433851161083276572159278385, −6.33936217767341830227216817971, −5.08644532893701287405730579469, −3.75521121491802393346092585118, −2.77031961470767503872629884662, 1.57218322916192363344311116267, 3.12918914925627571539520560095, 4.92680694316637013227842401028, 6.12897633470547152576738752297, 6.74677388518014076294327595665, 8.101475624121643854016792628220, 9.141405200033575166346468114886, 9.553639625601077482487287119231, 11.29382567476971384000956035775, 12.01921646940818089154177397630

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