Properties

Label 2-288-72.67-c0-0-0
Degree $2$
Conductor $288$
Sign $0.766 - 0.642i$
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  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 17-s + 19-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + (−0.5 − 0.866i)51-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)59-s + (−0.5 + 0.866i)67-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 17-s + 19-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + (−0.5 − 0.866i)51-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)59-s + (−0.5 + 0.866i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(0.143730\)
Root analytic conductor: \(0.379118\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8491845197\)
\(L(\frac12)\) \(\approx\) \(0.8491845197\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
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

−11.97573815334376466413570916088, −11.01712646271616680217059782019, −10.30123294405171983945319577700, −9.265452115718542584533803237753, −8.492488601413409686406975514725, −7.52059976324434673095561067634, −6.02084406712930474359488698716, −4.94752894790484327388806822254, −3.76292847751230084723448605318, −2.55740197511479142388327182984, 1.88464288874454619446146495074, 3.23161892249714255438115202390, 4.80648808073003878039043101676, 6.17276682240831828685795082405, 7.22493652639742857624836174268, 7.917735007051806506900779299386, 9.063727139643326893947662893369, 9.868532226469307474670523599663, 11.20665783989842032015504872554, 12.01611523951067512914040256204

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