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Properties

Label 2-288-32.5-c1-0-6
Degree $2$
Conductor $288$
Sign $-0.195 - 0.980i$
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  + 1.41i·2-s − 2.00·4-s + (3.12 + 1.29i)5-s + (1 + i)7-s − 2.82i·8-s + (−1.82 + 4.41i)10-s + (−0.121 + 0.292i)11-s + (1.70 − 0.707i)13-s + (−1.41 + 1.41i)14-s + 4.00·16-s + 2.82i·17-s + (−5.53 + 2.29i)19-s + (−6.24 − 2.58i)20-s + (−0.414 − 0.171i)22-s + (−0.171 + 0.171i)23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + (1.39 + 0.578i)5-s + (0.377 + 0.377i)7-s − 1.00i·8-s + (−0.578 + 1.39i)10-s + (−0.0365 + 0.0883i)11-s + (0.473 − 0.196i)13-s + (−0.377 + 0.377i)14-s + 1.00·16-s + 0.685i·17-s + (−1.26 + 0.526i)19-s + (−1.39 − 0.578i)20-s + (−0.0883 − 0.0365i)22-s + (−0.0357 + 0.0357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.195 - 0.980i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923361 + 1.12511i\)
\(L(\frac12)\) \(\approx\) \(0.923361 + 1.12511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
good5 \( 1 + (-3.12 - 1.29i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (0.121 - 0.292i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.70 + 0.707i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + (5.53 - 2.29i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.171 - 0.171i)T - 23iT^{2} \)
29 \( 1 + (1.12 + 2.70i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.70 - 0.707i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.82 + 5.82i)T - 41iT^{2} \)
43 \( 1 + (-3.29 + 7.94i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (3.12 - 7.53i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-6.12 - 2.53i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.292 - 0.707i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-1.53 - 3.70i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-0.171 - 0.171i)T + 71iT^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (6.12 - 2.53i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.65 - 2.65i)T + 89iT^{2} \)
97 \( 1 + 1.51T + 97T^{2} \)
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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

−12.39123074006890050180457752772, −10.78561252152661652061368888026, −10.11000884382437960037950330715, −9.072703227549159678625772114601, −8.276112455032660916358225904149, −7.00672950726704440699684843683, −6.03131193345928915834674960077, −5.49396425573329848786589786716, −3.94619168554748693802608659141, −2.06312245453306001494040830920, 1.32544167964676166163749752552, 2.56908370604028217141744108294, 4.28166331738497335254708313569, 5.23564127947105950268077066049, 6.34542524003591990203514964658, 8.023693898639926356864332301895, 9.119294538604613029533580422915, 9.598158330568536985125485185625, 10.74199012980585965385647669735, 11.30588606354491120463334162617

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