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Properties

Label 2-288-96.83-c1-0-8
Degree $2$
Conductor $288$
Sign $0.970 + 0.239i$
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  + (1.24 − 0.673i)2-s + (1.09 − 1.67i)4-s + (1.11 + 2.70i)5-s + (1.57 + 1.57i)7-s + (0.232 − 2.81i)8-s + (3.21 + 2.60i)10-s + (1.00 + 2.41i)11-s + (−6.48 − 2.68i)13-s + (3.01 + 0.896i)14-s + (−1.60 − 3.66i)16-s − 0.520·17-s + (2.95 − 7.14i)19-s + (5.75 + 1.08i)20-s + (2.87 + 2.33i)22-s + (1.35 + 1.35i)23-s + ⋯
L(s)  = 1  + (0.879 − 0.476i)2-s + (0.546 − 0.837i)4-s + (0.500 + 1.20i)5-s + (0.593 + 0.593i)7-s + (0.0823 − 0.996i)8-s + (1.01 + 0.824i)10-s + (0.301 + 0.729i)11-s + (−1.79 − 0.745i)13-s + (0.805 + 0.239i)14-s + (−0.401 − 0.915i)16-s − 0.126·17-s + (0.678 − 1.63i)19-s + (1.28 + 0.241i)20-s + (0.612 + 0.497i)22-s + (0.282 + 0.282i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22589 - 0.270520i\)
\(L(\frac12)\) \(\approx\) \(2.22589 - 0.270520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 + 0.673i)T \)
3 \( 1 \)
good5 \( 1 + (-1.11 - 2.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.57 - 1.57i)T + 7iT^{2} \)
11 \( 1 + (-1.00 - 2.41i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (6.48 + 2.68i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.520T + 17T^{2} \)
19 \( 1 + (-2.95 + 7.14i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.35 - 1.35i)T + 23iT^{2} \)
29 \( 1 + (5.31 + 2.20i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.54iT - 31T^{2} \)
37 \( 1 + (3.79 - 1.57i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.08 - 1.08i)T - 41iT^{2} \)
43 \( 1 + (2.71 - 1.12i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + (-3.70 + 1.53i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.19 - 1.32i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.78 - 9.12i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-10.9 - 4.53i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-6.83 + 6.83i)T - 71iT^{2} \)
73 \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \)
79 \( 1 - 8.79T + 79T^{2} \)
83 \( 1 + (13.9 + 5.79i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.09 + 7.09i)T + 89iT^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−11.73720002143617947137058200449, −11.04993062399712250714129426962, −10.03087550588354164698120586594, −9.397591398315869332175203719149, −7.46656762538692360263974522882, −6.81868059540340921173066137787, −5.51497093057877368773835352070, −4.70593402562860941742098640385, −2.95943246518352128220267562024, −2.20526491410463918858127644412, 1.83490494024512346110651977321, 3.75447653745978775766897524278, 4.90092776687140332075303590814, 5.50154352064201928686322866180, 6.89765923183909353660777114069, 7.85632012133259147344232287526, 8.820668067879722269560868205603, 9.890473981845491579289495007260, 11.22049295326007914225463326368, 12.19411291262118875422040053405

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