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Properties

Label 2-288-9.4-c3-0-12
Degree $2$
Conductor $288$
Sign $0.877 - 0.478i$
Analytic cond. $16.9925$
Root an. cond. $4.12220$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.73 + 2.13i)3-s + (−3.14 + 5.44i)5-s + (−6.02 − 10.4i)7-s + (17.8 − 20.2i)9-s + (−12.0 − 20.8i)11-s + (−4.97 + 8.62i)13-s + (3.23 − 32.4i)15-s + 39.3·17-s − 45.0·19-s + (50.8 + 36.5i)21-s + (54.7 − 94.7i)23-s + (42.7 + 74.0i)25-s + (−41.1 + 134. i)27-s + (127. + 221. i)29-s + (−24.3 + 42.1i)31-s + ⋯
L(s)  = 1  + (−0.911 + 0.411i)3-s + (−0.280 + 0.486i)5-s + (−0.325 − 0.563i)7-s + (0.661 − 0.750i)9-s + (−0.329 − 0.570i)11-s + (−0.106 + 0.183i)13-s + (0.0557 − 0.559i)15-s + 0.561·17-s − 0.543·19-s + (0.528 + 0.379i)21-s + (0.496 − 0.859i)23-s + (0.342 + 0.592i)25-s + (−0.293 + 0.955i)27-s + (0.817 + 1.41i)29-s + (−0.141 + 0.244i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.877 - 0.478i$
Analytic conductor: \(16.9925\)
Root analytic conductor: \(4.12220\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :3/2),\ 0.877 - 0.478i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.077096169\)
\(L(\frac12)\) \(\approx\) \(1.077096169\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.73 - 2.13i)T \)
good5 \( 1 + (3.14 - 5.44i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (6.02 + 10.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (12.0 + 20.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (4.97 - 8.62i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 39.3T + 4.91e3T^{2} \)
19 \( 1 + 45.0T + 6.85e3T^{2} \)
23 \( 1 + (-54.7 + 94.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-127. - 221. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (24.3 - 42.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 173.T + 5.06e4T^{2} \)
41 \( 1 + (-104. + 180. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-194. - 336. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-76.5 - 132. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 133.T + 1.48e5T^{2} \)
59 \( 1 + (-267. + 463. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-327. - 566. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-13.9 + 24.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 49.2T + 3.57e5T^{2} \)
73 \( 1 - 188.T + 3.89e5T^{2} \)
79 \( 1 + (-562. - 973. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (540. + 936. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 338.T + 7.04e5T^{2} \)
97 \( 1 + (656. + 1.13e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.14179109188739244901375946709, −10.72347901188478551970794290573, −9.840000922087864575798699570302, −8.659776837750978335894123757233, −7.28011926282116092118378631741, −6.54013220406945262423261343577, −5.40590322379688772384913668722, −4.24445913601410517689180009969, −3.09029087050670082135558277333, −0.811303668366828307901883998759, 0.71821154927636369175737450052, 2.39032339561549141898302436249, 4.27187418410205787155562551894, 5.32421326377437754660467972292, 6.22341516031720709475327569229, 7.37237266298107367414476225222, 8.269794993806298473504392300553, 9.540292350849889834784769958690, 10.41902633581905108432838784765, 11.50156518711405933530392431057

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