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Properties

Label 2-288-288.83-c1-0-24
Degree $2$
Conductor $288$
Sign $0.655 - 0.755i$
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  + (−0.447 + 1.34i)2-s + (−0.897 + 1.48i)3-s + (−1.59 − 1.20i)4-s + (1.68 − 2.19i)5-s + (−1.58 − 1.86i)6-s + (3.57 − 0.957i)7-s + (2.32 − 1.60i)8-s + (−1.38 − 2.65i)9-s + (2.19 + 3.24i)10-s + (2.76 + 0.364i)11-s + (3.21 − 1.29i)12-s + (−0.0949 − 0.721i)13-s + (−0.315 + 5.22i)14-s + (1.74 + 4.47i)15-s + (1.11 + 3.84i)16-s − 4.88·17-s + ⋯
L(s)  = 1  + (−0.316 + 0.948i)2-s + (−0.518 + 0.855i)3-s + (−0.799 − 0.600i)4-s + (0.754 − 0.983i)5-s + (−0.647 − 0.762i)6-s + (1.35 − 0.361i)7-s + (0.822 − 0.568i)8-s + (−0.463 − 0.886i)9-s + (0.694 + 1.02i)10-s + (0.834 + 0.109i)11-s + (0.927 − 0.372i)12-s + (−0.0263 − 0.200i)13-s + (−0.0842 + 1.39i)14-s + (0.450 + 1.15i)15-s + (0.278 + 0.960i)16-s − 1.18·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.655 - 0.755i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01334 + 0.462517i\)
\(L(\frac12)\) \(\approx\) \(1.01334 + 0.462517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.447 - 1.34i)T \)
3 \( 1 + (0.897 - 1.48i)T \)
good5 \( 1 + (-1.68 + 2.19i)T + (-1.29 - 4.82i)T^{2} \)
7 \( 1 + (-3.57 + 0.957i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.76 - 0.364i)T + (10.6 + 2.84i)T^{2} \)
13 \( 1 + (0.0949 + 0.721i)T + (-12.5 + 3.36i)T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 + (-3.20 + 7.72i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.38 - 8.91i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.969 - 0.743i)T + (7.50 - 28.0i)T^{2} \)
31 \( 1 + (-2.36 + 1.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.41 + 1.82i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.884 - 0.237i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.36 - 10.3i)T + (-41.5 - 11.1i)T^{2} \)
47 \( 1 + (1.50 + 0.868i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.81 + 1.99i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.40 - 2.61i)T + (15.2 + 56.9i)T^{2} \)
61 \( 1 + (3.79 + 4.93i)T + (-15.7 + 58.9i)T^{2} \)
67 \( 1 + (0.814 + 6.18i)T + (-64.7 + 17.3i)T^{2} \)
71 \( 1 + (6.73 - 6.73i)T - 71iT^{2} \)
73 \( 1 + (-0.311 - 0.311i)T + 73iT^{2} \)
79 \( 1 + (1.58 - 2.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.50 - 4.98i)T + (21.4 - 80.1i)T^{2} \)
89 \( 1 + (1.28 + 1.28i)T + 89iT^{2} \)
97 \( 1 + (2.19 - 3.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.60027060295337411085904807348, −11.00312080158034617958923029256, −9.563015812892212787493344835649, −9.281862072453864087411172800842, −8.257852466435556931246271115932, −6.98848771177971753026207727362, −5.74064572048403477388749091397, −4.94094066873035236935785553892, −4.29531928964979107645083663758, −1.24072878882794454063161760012, 1.61343048033590551471345111873, 2.47639788233897158083333196535, 4.36105718518996766651162168819, 5.72559326546335332923212260852, 6.78023724185214390524136017445, 7.992796309854867345693230507198, 8.784766248826629209329805662275, 10.19771159784475533982543892932, 10.82524876247068533082505792469, 11.73186671509414581699224263121

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