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Properties

Label 2-288-288.11-c1-0-3
Degree $2$
Conductor $288$
Sign $0.410 - 0.911i$
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  + (−1.34 − 0.444i)2-s + (−0.318 − 1.70i)3-s + (1.60 + 1.19i)4-s + (0.201 + 1.53i)5-s + (−0.330 + 2.42i)6-s + (0.765 + 2.85i)7-s + (−1.62 − 2.31i)8-s + (−2.79 + 1.08i)9-s + (0.410 − 2.14i)10-s + (−2.13 + 1.63i)11-s + (1.52 − 3.11i)12-s + (−3.96 + 5.16i)13-s + (0.242 − 4.17i)14-s + (2.54 − 0.831i)15-s + (1.14 + 3.83i)16-s − 4.99·17-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)2-s + (−0.183 − 0.982i)3-s + (0.802 + 0.597i)4-s + (0.0902 + 0.685i)5-s + (−0.134 + 0.990i)6-s + (0.289 + 1.07i)7-s + (−0.573 − 0.819i)8-s + (−0.932 + 0.361i)9-s + (0.129 − 0.679i)10-s + (−0.642 + 0.493i)11-s + (0.439 − 0.898i)12-s + (−1.09 + 1.43i)13-s + (0.0649 − 1.11i)14-s + (0.657 − 0.214i)15-s + (0.287 + 0.957i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.410 - 0.911i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473491 + 0.306170i\)
\(L(\frac12)\) \(\approx\) \(0.473491 + 0.306170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.444i)T \)
3 \( 1 + (0.318 + 1.70i)T \)
good5 \( 1 + (-0.201 - 1.53i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.765 - 2.85i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.13 - 1.63i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (3.96 - 5.16i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 4.99T + 17T^{2} \)
19 \( 1 + (-1.11 + 0.460i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.75 - 1.54i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.24 + 0.295i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (-6.12 + 3.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.82 - 6.82i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.24 - 4.65i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.77 + 6.22i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (5.05 + 2.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.936 + 2.26i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-9.62 + 1.26i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (-0.825 + 6.27i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (6.82 - 8.89i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-11.1 - 11.1i)T + 71iT^{2} \)
73 \( 1 + (-4.15 + 4.15i)T - 73iT^{2} \)
79 \( 1 + (-2.07 + 3.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.51 + 0.199i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (6.21 - 6.21i)T - 89iT^{2} \)
97 \( 1 + (0.292 - 0.506i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.64609817373870511800151773161, −11.36751747588794456686317338093, −10.06094000068060674964518686575, −9.050121605744820896865228480944, −8.219080223023710117068533424706, −6.99042830530602758607501255490, −6.66565657186158493862404975394, −5.05135692359306285268619491331, −2.69786086666054244109453091100, −2.03449274385142002376799440264, 0.55470401506868492317206755722, 2.93864195026203263294643072396, 4.75135168697964329695449088831, 5.43511959288960181854414291573, 6.92012423292227862428718188537, 8.016626180915818971223464203548, 8.817478248724746961009029759326, 9.798312835287520209890154850876, 10.64799165758515299802482746463, 11.02491439185989865609470420413

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