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Properties

Label 2-288-32.13-c1-0-16
Degree $2$
Conductor $288$
Sign $0.534 + 0.845i$
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  + (1.26 − 0.635i)2-s + (1.19 − 1.60i)4-s + (−0.707 + 0.292i)5-s + (1.68 − 1.68i)7-s + (0.484 − 2.78i)8-s + (−0.707 + 0.819i)10-s + (0.334 + 0.808i)11-s + (1.09 + 0.451i)13-s + (1.05 − 3.20i)14-s + (−1.15 − 3.82i)16-s − 0.224i·17-s + (−2.87 − 1.19i)19-s + (−0.372 + 1.48i)20-s + (0.936 + 0.808i)22-s + (3.68 + 3.68i)23-s + ⋯
L(s)  = 1  + (0.893 − 0.449i)2-s + (0.595 − 0.803i)4-s + (−0.316 + 0.130i)5-s + (0.637 − 0.637i)7-s + (0.171 − 0.985i)8-s + (−0.223 + 0.259i)10-s + (0.100 + 0.243i)11-s + (0.302 + 0.125i)13-s + (0.282 − 0.855i)14-s + (−0.289 − 0.957i)16-s − 0.0545i·17-s + (−0.660 − 0.273i)19-s + (−0.0832 + 0.332i)20-s + (0.199 + 0.172i)22-s + (0.768 + 0.768i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.534 + 0.845i$
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: \((2,\ 288,\ (\ :1/2),\ 0.534 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85011 - 1.01915i\)
\(L(\frac12)\) \(\approx\) \(1.85011 - 1.01915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.635i)T \)
3 \( 1 \)
good5 \( 1 + (0.707 - 0.292i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.68 + 1.68i)T - 7iT^{2} \)
11 \( 1 + (-0.334 - 0.808i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.09 - 0.451i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.224iT - 17T^{2} \)
19 \( 1 + (2.87 + 1.19i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.68 - 3.68i)T + 23iT^{2} \)
29 \( 1 + (2.34 - 5.66i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (9.87 - 4.09i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.37 + 6.37i)T + 41iT^{2} \)
43 \( 1 + (-1.90 - 4.60i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.542iT - 47T^{2} \)
53 \( 1 + (-3.91 - 9.46i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.36 + 1.39i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.398 + 0.962i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.48 + 3.57i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.39 + 5.39i)T - 71iT^{2} \)
73 \( 1 + (5.15 + 5.15i)T + 73iT^{2} \)
79 \( 1 - 8.39iT - 79T^{2} \)
83 \( 1 + (11.2 + 4.64i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.92 + 5.92i)T - 89iT^{2} \)
97 \( 1 + 4.19T + 97T^{2} \)
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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

−11.62623657558525012378174192828, −10.95193310771658222257212801191, −10.15494533923751553081421251391, −8.888343722881539145323852545707, −7.51122492722212942041580592647, −6.70025068528647324684396043386, −5.33663532597753138291687514291, −4.35962317638711544224783059959, −3.28938792433886968014354235951, −1.57178442016337962259938977379, 2.28983966640033982401584180165, 3.78717418656459702085449626104, 4.86895317854636072547163406695, 5.88280179636556615718495697817, 6.89220383681332021876611660587, 8.204028861624533727009862549879, 8.605316302625588970131547832662, 10.29611773955221307664533893711, 11.42693150705541497649961299189, 11.97684255491977478698686543602

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