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Properties

Label 2-288-72.61-c1-0-3
Degree $2$
Conductor $288$
Sign $0.999 - 0.00404i$
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  + (−1.69 − 0.378i)3-s + (−1.97 + 1.14i)5-s + (0.907 − 1.57i)7-s + (2.71 + 1.27i)9-s + (4.24 + 2.44i)11-s + (4.00 − 2.31i)13-s + (3.77 − 1.18i)15-s + 1.92·17-s − 2.12i·19-s + (−2.12 + 2.31i)21-s + (1.15 + 2.00i)23-s + (0.101 − 0.175i)25-s + (−4.10 − 3.18i)27-s + (3.16 + 1.82i)29-s + (2.65 + 4.60i)31-s + ⋯
L(s)  = 1  + (−0.975 − 0.218i)3-s + (−0.883 + 0.510i)5-s + (0.343 − 0.594i)7-s + (0.904 + 0.426i)9-s + (1.27 + 0.738i)11-s + (1.11 − 0.641i)13-s + (0.973 − 0.304i)15-s + 0.467·17-s − 0.488i·19-s + (−0.464 + 0.505i)21-s + (0.241 + 0.418i)23-s + (0.0203 − 0.0351i)25-s + (−0.789 − 0.613i)27-s + (0.587 + 0.339i)29-s + (0.477 + 0.826i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.999 - 0.00404i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.999 - 0.00404i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.947282 + 0.00191701i\)
\(L(\frac12)\) \(\approx\) \(0.947282 + 0.00191701i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.69 + 0.378i)T \)
good5 \( 1 + (1.97 - 1.14i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.907 + 1.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.24 - 2.44i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.00 + 2.31i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 + 2.12iT - 19T^{2} \)
23 \( 1 + (-1.15 - 2.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.16 - 1.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.65 - 4.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.98iT - 37T^{2} \)
41 \( 1 + (2.36 + 4.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 1.27i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.02 + 3.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.95iT - 53T^{2} \)
59 \( 1 + (3.05 - 1.76i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 - 0.991i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.72 + 4.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (-4.97 + 8.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.12 + 1.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.49T + 89T^{2} \)
97 \( 1 + (-6.99 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.74903513081232384980577511086, −10.95997883388610245238209632735, −10.33146493597989331169131007071, −8.941749347694586177262330294614, −7.57488449490463586578284561612, −7.02956387918430001768962744951, −5.93585286690072582647556337005, −4.55251883880488792887745266914, −3.60334639437120257930796101870, −1.19927079369655768694511600922, 1.15081119018580080315446501542, 3.72165571677942707758263082007, 4.56512901297958037958249383919, 5.88167018678048591374847225602, 6.62639440752653165496061715713, 8.144015492833417042220870770737, 8.852683221896724144330502513024, 10.00497716776796284404440523587, 11.33103022874811849223567649398, 11.65142963139202038543344476846

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