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Properties

Label 2-288-72.13-c1-0-2
Degree $2$
Conductor $288$
Sign $0.178 - 0.983i$
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  + (−0.294 + 1.70i)3-s + (3.17 + 1.83i)5-s + (0.191 + 0.332i)7-s + (−2.82 − 1.00i)9-s + (1.73 − 1.00i)11-s + (−0.397 − 0.229i)13-s + (−4.06 + 4.87i)15-s − 4.08·17-s + 4.72i·19-s + (−0.623 + 0.229i)21-s + (2.97 − 5.15i)23-s + (4.21 + 7.29i)25-s + (2.54 − 4.52i)27-s + (−2.03 + 1.17i)29-s + (−0.592 + 1.02i)31-s + ⋯
L(s)  = 1  + (−0.170 + 0.985i)3-s + (1.41 + 0.819i)5-s + (0.0725 + 0.125i)7-s + (−0.942 − 0.335i)9-s + (0.524 − 0.302i)11-s + (−0.110 − 0.0636i)13-s + (−1.04 + 1.25i)15-s − 0.990·17-s + 1.08i·19-s + (−0.136 + 0.0501i)21-s + (0.620 − 1.07i)23-s + (0.842 + 1.45i)25-s + (0.490 − 0.871i)27-s + (−0.378 + 0.218i)29-s + (−0.106 + 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.178 - 0.983i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.178 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12392 + 0.938249i\)
\(L(\frac12)\) \(\approx\) \(1.12392 + 0.938249i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.294 - 1.70i)T \)
good5 \( 1 + (-3.17 - 1.83i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.191 - 0.332i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + 1.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.397 + 0.229i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.08T + 17T^{2} \)
19 \( 1 - 4.72iT - 19T^{2} \)
23 \( 1 + (-2.97 + 5.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.03 - 1.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.592 - 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.74iT - 37T^{2} \)
41 \( 1 + (-4.75 + 8.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.03 - 0.598i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.27 + 5.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.63iT - 53T^{2} \)
59 \( 1 + (0.603 + 0.348i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.23 + 2.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.87 + 5.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.49 + 3.16i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.56T + 89T^{2} \)
97 \( 1 + (2.98 + 5.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.77274540731550969779435878210, −10.77483274351725087419154350520, −10.24411441458639241397854443161, −9.354789581584695532402971665095, −8.555108995799569297619428893147, −6.76803656865534996867033286024, −6.02865989944431764296982060361, −5.02774290524843423328287053381, −3.57727837488218094515359973097, −2.25657690784872211881180817696, 1.30216382549038895079581597462, 2.46779689445587613088173551942, 4.66833687839899861867582991461, 5.71826947885520168081071419292, 6.57657877665536139118566848070, 7.60518365758036553366318185087, 9.032018630676711038893246228354, 9.337599045849106781842543510735, 10.80710289896569817870412683794, 11.69931665762736430808395752946

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