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Properties

Label 2-288-36.23-c1-0-6
Degree $2$
Conductor $288$
Sign $0.981 - 0.190i$
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  + (1.71 + 0.211i)3-s + (−0.135 − 0.0779i)5-s + (−0.349 + 0.201i)7-s + (2.91 + 0.726i)9-s + (2.28 + 3.95i)11-s + (2.14 − 3.71i)13-s + (−0.215 − 0.162i)15-s − 2.84i·17-s + 0.958i·19-s + (−0.643 + 0.272i)21-s + (−2.71 + 4.69i)23-s + (−2.48 − 4.30i)25-s + (4.85 + 1.86i)27-s + (−4.88 + 2.82i)29-s + (−8.53 − 4.92i)31-s + ⋯
L(s)  = 1  + (0.992 + 0.122i)3-s + (−0.0603 − 0.0348i)5-s + (−0.132 + 0.0762i)7-s + (0.970 + 0.242i)9-s + (0.689 + 1.19i)11-s + (0.594 − 1.02i)13-s + (−0.0556 − 0.0419i)15-s − 0.688i·17-s + 0.219i·19-s + (−0.140 + 0.0595i)21-s + (−0.565 + 0.979i)23-s + (−0.497 − 0.861i)25-s + (0.933 + 0.358i)27-s + (−0.907 + 0.524i)29-s + (−1.53 − 0.885i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.981 - 0.190i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76460 + 0.169592i\)
\(L(\frac12)\) \(\approx\) \(1.76460 + 0.169592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.211i)T \)
good5 \( 1 + (0.135 + 0.0779i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.349 - 0.201i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.28 - 3.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.14 + 3.71i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.84iT - 17T^{2} \)
19 \( 1 - 0.958iT - 19T^{2} \)
23 \( 1 + (2.71 - 4.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.88 - 2.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.53 + 4.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.89T + 37T^{2} \)
41 \( 1 + (7.79 + 4.49i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.59 + 1.49i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.22 + 2.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.45iT - 53T^{2} \)
59 \( 1 + (1.18 - 2.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.10 + 7.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.4 + 6.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 4.53T + 73T^{2} \)
79 \( 1 + (-2.84 + 1.64i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.812 - 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.45iT - 89T^{2} \)
97 \( 1 + (0.162 + 0.281i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.99144550892970523654195864567, −10.75217246599941091788075172487, −9.689726063076643896160326102082, −9.177526818088072246741873392944, −7.900650592986565964846756856037, −7.28626854225475890133739421967, −5.85081825374068216767167630930, −4.39449992980074791325414084511, −3.37108040352370386290718957996, −1.88033195889449582747437391617, 1.69962417788745087065657183037, 3.36268021973935638572442232861, 4.18426821613233218180807582851, 6.00137983199272111686000701250, 6.92548434583806033035531273470, 8.138011089504679319304783891138, 8.873351678832272506042880193278, 9.643321483786349683347424324980, 10.90794823859126994634575759423, 11.71508987238288197535418084621

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