Properties

Label 2-288-288.11-c1-0-31
Degree $2$
Conductor $288$
Sign $-0.220 + 0.975i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.370 − 1.36i)2-s + (1.60 + 0.661i)3-s + (−1.72 + 1.01i)4-s + (−0.155 − 1.18i)5-s + (0.309 − 2.42i)6-s + (−1.23 − 4.62i)7-s + (2.01 + 1.98i)8-s + (2.12 + 2.11i)9-s + (−1.55 + 0.651i)10-s + (0.714 − 0.548i)11-s + (−3.43 + 0.476i)12-s + (1.55 − 2.02i)13-s + (−5.85 + 3.40i)14-s + (0.533 − 1.99i)15-s + (1.95 − 3.48i)16-s − 5.41·17-s + ⋯
L(s)  = 1  + (−0.261 − 0.965i)2-s + (0.924 + 0.381i)3-s + (−0.862 + 0.505i)4-s + (−0.0697 − 0.529i)5-s + (0.126 − 0.991i)6-s + (−0.468 − 1.74i)7-s + (0.713 + 0.700i)8-s + (0.708 + 0.705i)9-s + (−0.493 + 0.205i)10-s + (0.215 − 0.165i)11-s + (−0.990 + 0.137i)12-s + (0.430 − 0.561i)13-s + (−1.56 + 0.910i)14-s + (0.137 − 0.516i)15-s + (0.489 − 0.872i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.220 + 0.975i$
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: \((2,\ 288,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828513 - 1.03719i\)
\(L(\frac12)\) \(\approx\) \(0.828513 - 1.03719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.370 + 1.36i)T \)
3 \( 1 + (-1.60 - 0.661i)T \)
good5 \( 1 + (0.155 + 1.18i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (1.23 + 4.62i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.714 + 0.548i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-1.55 + 2.02i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + (-3.09 + 1.28i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.48 - 1.20i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.97 + 0.523i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (1.91 - 1.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.235 + 0.568i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.92 - 10.9i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.771 - 1.00i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (-4.59 - 2.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.55 - 6.16i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-5.22 + 0.688i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (1.75 - 13.3i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-4.89 + 6.37i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-6.14 - 6.14i)T + 71iT^{2} \)
73 \( 1 + (-8.86 + 8.86i)T - 73iT^{2} \)
79 \( 1 + (-2.86 + 4.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.89 + 0.644i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (-3.18 + 3.18i)T - 89iT^{2} \)
97 \( 1 + (6.48 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.13929285382866478614470056034, −10.64866794114653268588569168660, −9.632463001908986589060666202023, −8.965387492335587666393825465235, −7.949648495773138246500414609003, −7.02069999393868294260283213332, −4.81776879924753404822343531157, −3.97231817822834177780412740299, −3.02597742398423268115407580394, −1.11430976796082015256111456218, 2.18477655482071171006196511407, 3.63966725172297540354732704777, 5.28073033748045121845421597793, 6.50563237999505503404269820764, 7.06537371298542317629653694928, 8.431582515394669467193741573369, 9.014030000218348678053442979997, 9.599918579078443981437212905045, 11.09386802351437722914627959805, 12.37721667718841912651841304146

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