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Properties

Label 2-288-32.13-c1-0-1
Degree $2$
Conductor $288$
Sign $-0.985 - 0.170i$
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  + (0.443 + 1.34i)2-s + (−1.60 + 1.19i)4-s + (−0.707 + 0.292i)5-s + (−2.27 + 2.27i)7-s + (−2.31 − 1.62i)8-s + (−0.707 − 0.819i)10-s + (1.49 + 3.60i)11-s + (−4.50 − 1.86i)13-s + (−4.05 − 2.04i)14-s + (1.15 − 3.82i)16-s + 3.05i·17-s + (3.87 + 1.60i)19-s + (0.786 − 1.31i)20-s + (−4.17 + 3.60i)22-s + (−0.271 − 0.271i)23-s + ⋯
L(s)  = 1  + (0.313 + 0.949i)2-s + (−0.803 + 0.595i)4-s + (−0.316 + 0.130i)5-s + (−0.858 + 0.858i)7-s + (−0.817 − 0.575i)8-s + (−0.223 − 0.259i)10-s + (0.450 + 1.08i)11-s + (−1.24 − 0.517i)13-s + (−1.08 − 0.545i)14-s + (0.289 − 0.957i)16-s + 0.740i·17-s + (0.889 + 0.368i)19-s + (0.175 − 0.293i)20-s + (−0.891 + 0.768i)22-s + (−0.0565 − 0.0565i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.985 - 0.170i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.985 - 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0795327 + 0.927458i\)
\(L(\frac12)\) \(\approx\) \(0.0795327 + 0.927458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.443 - 1.34i)T \)
3 \( 1 \)
good5 \( 1 + (0.707 - 0.292i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.27 - 2.27i)T - 7iT^{2} \)
11 \( 1 + (-1.49 - 3.60i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (4.50 + 1.86i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.05iT - 17T^{2} \)
19 \( 1 + (-3.87 - 1.60i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.271 + 0.271i)T + 23iT^{2} \)
29 \( 1 + (-0.931 + 2.24i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (-3.63 + 1.50i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.54 - 1.54i)T + 41iT^{2} \)
43 \( 1 + (-0.748 - 1.80i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.37iT - 47T^{2} \)
53 \( 1 + (1.67 + 4.04i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-10.1 + 4.19i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.35 + 3.28i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.99 - 4.81i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (6.47 - 6.47i)T - 71iT^{2} \)
73 \( 1 + (2.84 + 2.84i)T + 73iT^{2} \)
79 \( 1 - 9.74iT - 79T^{2} \)
83 \( 1 + (-9.04 - 3.74i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.58 - 7.58i)T - 89iT^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
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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

−12.42376763878702419926189724513, −11.74678193654262447681552094005, −9.851407038335265860328072001405, −9.532037232718064066092828188361, −8.175414674059395450498188446646, −7.34266264965008645608080866138, −6.35303908114638384991771052407, −5.36696333216412382689019201068, −4.16848376339785464784552607239, −2.81787189216350794590056320739, 0.63488933654294328339705217403, 2.78360642927048567175719105136, 3.85022350294873571597673763035, 4.92530809389962668758024677650, 6.29464993459815772109495106849, 7.45119847384561073704605545344, 8.836472091121579397868342921255, 9.715148326295724964435359285318, 10.40402578394823338845232158110, 11.66961853125269159488523416398

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