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Properties

Label 2-288-72.61-c1-0-9
Degree $2$
Conductor $288$
Sign $0.0871 + 0.996i$
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  + (0.866 − 1.5i)3-s + (−1.73 + i)5-s + (2 − 3.46i)7-s + (−1.5 − 2.59i)9-s + (−2.59 − 1.5i)11-s + (1.73 − i)13-s + 3.46i·15-s + 5·17-s − i·19-s + (−3.46 − 6i)21-s + (1 + 1.73i)23-s + (−0.500 + 0.866i)25-s − 5.19·27-s + (−2 − 3.46i)31-s + (−4.5 + 2.59i)33-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (−0.774 + 0.447i)5-s + (0.755 − 1.30i)7-s + (−0.5 − 0.866i)9-s + (−0.783 − 0.452i)11-s + (0.480 − 0.277i)13-s + 0.894i·15-s + 1.21·17-s − 0.229i·19-s + (−0.755 − 1.30i)21-s + (0.208 + 0.361i)23-s + (−0.100 + 0.173i)25-s − 1.00·27-s + (−0.359 − 0.622i)31-s + (−0.783 + 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.0871 + 0.996i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.0871 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991199 - 0.908266i\)
\(L(\frac12)\) \(\approx\) \(0.991199 - 0.908266i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 1.5i)T \)
good5 \( 1 + (1.73 - i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.52 - 5.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 - 6i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.38854046637537668637854878810, −10.97435059037931186287363793222, −9.739351641446992412761709088026, −8.211342447869230299336353052980, −7.76504140347977063934648337782, −7.07298204724172767356251225288, −5.66957857793815988515310100919, −4.04041462531186537064021921306, −3.01510883609160373228349786580, −1.05769419556521366685285669188, 2.29619938113364418040097790512, 3.72200493391361527989947976254, 4.90963042318435002805189605630, 5.63498325059582975640730499846, 7.60548936776556772389054915728, 8.364420091481086857033298682894, 8.995635016008062883497132754434, 10.12288534609615861124136113393, 11.10338207124188628977013721883, 12.00944708124300443661432958917

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