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Properties

Label 2-288-9.4-c1-0-7
Degree $2$
Conductor $288$
Sign $0.939 + 0.342i$
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  + 1.73·3-s + (0.5 − 0.866i)5-s + (−0.866 − 1.5i)7-s + 2.99·9-s + (0.866 + 1.5i)11-s + (1.5 − 2.59i)13-s + (0.866 − 1.49i)15-s + 4·17-s − 6.92·19-s + (−1.49 − 2.59i)21-s + (−4.33 + 7.5i)23-s + (2 + 3.46i)25-s + 5.19·27-s + (−0.5 − 0.866i)29-s + (2.59 − 4.5i)31-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.223 − 0.387i)5-s + (−0.327 − 0.566i)7-s + 0.999·9-s + (0.261 + 0.452i)11-s + (0.416 − 0.720i)13-s + (0.223 − 0.387i)15-s + 0.970·17-s − 1.58·19-s + (−0.327 − 0.566i)21-s + (−0.902 + 1.56i)23-s + (0.400 + 0.692i)25-s + 1.00·27-s + (−0.0928 − 0.160i)29-s + (0.466 − 0.808i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76352 - 0.310957i\)
\(L(\frac12)\) \(\approx\) \(1.76352 - 0.310957i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.866 + 1.5i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + (4.33 - 7.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.59 + 4.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.33 + 7.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.33 - 7.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (2.59 + 4.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.33 + 7.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1.5 - 2.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.93877803211909878850875742400, −10.47945914245382278951178111711, −9.866625273952441897252578829499, −8.894796861241368213917043037564, −7.968035292621158324564347513473, −7.09287577731429003498621600581, −5.76447037055199073739771205970, −4.27994393804243794845438353251, −3.29732349864558400410302549750, −1.62855886492892848033720070546, 2.04491374239691159560657508922, 3.23941424425455232015458113527, 4.43851920392037031611869719722, 6.14854448922832309968041287027, 6.88519037871416508291482998912, 8.425319170336541108963226923956, 8.727328208055644810912531474251, 9.999738893605773267191415102720, 10.65365279046174988996094072909, 12.11260801201745828636281934241

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