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Properties

Label 2-288-96.35-c1-0-10
Degree $2$
Conductor $288$
Sign $0.428 + 0.903i$
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  + (−0.430 + 1.34i)2-s + (−1.62 − 1.15i)4-s + (−1.78 + 0.741i)5-s + (−2.33 − 2.33i)7-s + (2.26 − 1.69i)8-s + (−0.228 − 2.72i)10-s + (0.683 − 0.283i)11-s + (2.43 − 5.88i)13-s + (4.15 − 2.14i)14-s + (1.30 + 3.77i)16-s − 0.967·17-s + (−2.52 − 1.04i)19-s + (3.77 + 0.867i)20-s + (0.0871 + 1.04i)22-s + (−5.00 − 5.00i)23-s + ⋯
L(s)  = 1  + (−0.304 + 0.952i)2-s + (−0.814 − 0.579i)4-s + (−0.800 + 0.331i)5-s + (−0.883 − 0.883i)7-s + (0.800 − 0.599i)8-s + (−0.0721 − 0.863i)10-s + (0.206 − 0.0853i)11-s + (0.675 − 1.63i)13-s + (1.11 − 0.572i)14-s + (0.327 + 0.944i)16-s − 0.234·17-s + (−0.579 − 0.240i)19-s + (0.844 + 0.194i)20-s + (0.0185 + 0.222i)22-s + (−1.04 − 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.428 + 0.903i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.428 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408858 - 0.258476i\)
\(L(\frac12)\) \(\approx\) \(0.408858 - 0.258476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.430 - 1.34i)T \)
3 \( 1 \)
good5 \( 1 + (1.78 - 0.741i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.33 + 2.33i)T + 7iT^{2} \)
11 \( 1 + (-0.683 + 0.283i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.43 + 5.88i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.967T + 17T^{2} \)
19 \( 1 + (2.52 + 1.04i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.00 + 5.00i)T + 23iT^{2} \)
29 \( 1 + (-0.563 + 1.36i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.28iT - 31T^{2} \)
37 \( 1 + (-3.26 - 7.88i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.80 - 6.80i)T - 41iT^{2} \)
43 \( 1 + (-1.36 - 3.29i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.69iT - 47T^{2} \)
53 \( 1 + (1.85 + 4.47i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.05 + 9.79i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-10.8 - 4.48i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-1.49 + 3.60i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (9.85 - 9.85i)T - 71iT^{2} \)
73 \( 1 + (4.81 + 4.81i)T + 73iT^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (-1.15 + 2.77i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.82 - 6.82i)T + 89iT^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−11.48123816941284903611912352724, −10.39973549097243622049336003141, −9.859188007963605054740779984948, −8.390724927211522013290331982010, −7.83698220005613956593782380935, −6.73249871520991733825716002407, −5.99587670052260139749601639905, −4.40776849605883775074844803483, −3.41622888983582954923463434390, −0.40128733059905441254962069317, 1.94964197721109173273417931567, 3.53986046709343561932309658070, 4.37202099483102634664353713019, 5.96329480242703700528042896880, 7.29406286497518445538544121158, 8.664989870443837254328188146820, 9.022556444466087481506415843094, 10.08466973696879536408496787363, 11.25304814002176929236673321603, 12.00224382506100859309184562508

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