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Properties

Label 2-288-9.4-c1-0-8
Degree $2$
Conductor $288$
Sign $0.450 + 0.892i$
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  + (0.637 − 1.61i)3-s + (1.68 − 2.92i)5-s + (2.35 + 4.07i)7-s + (−2.18 − 2.05i)9-s + (0.437 + 0.758i)11-s + (0.686 − 1.18i)13-s + (−3.62 − 4.57i)15-s − 2.37·17-s − 5.57·19-s + (8.05 − 1.18i)21-s + (2.35 − 4.07i)23-s + (−3.18 − 5.51i)25-s + (−4.70 + 2.20i)27-s + (2.68 + 4.65i)29-s + (−3.22 + 5.58i)31-s + ⋯
L(s)  = 1  + (0.368 − 0.929i)3-s + (0.754 − 1.30i)5-s + (0.888 + 1.53i)7-s + (−0.728 − 0.684i)9-s + (0.131 + 0.228i)11-s + (0.190 − 0.329i)13-s + (−0.936 − 1.18i)15-s − 0.575·17-s − 1.27·19-s + (1.75 − 0.259i)21-s + (0.490 − 0.849i)23-s + (−0.637 − 1.10i)25-s + (−0.905 + 0.425i)27-s + (0.498 + 0.863i)29-s + (−0.579 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.450 + 0.892i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.450 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41255 - 0.869374i\)
\(L(\frac12)\) \(\approx\) \(1.41255 - 0.869374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.637 + 1.61i)T \)
good5 \( 1 + (-1.68 + 2.92i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.35 - 4.07i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.437 - 0.758i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.686 + 1.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.37T + 17T^{2} \)
19 \( 1 + 5.57T + 19T^{2} \)
23 \( 1 + (-2.35 + 4.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.437 + 0.758i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.35 - 4.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-4.26 + 7.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.05 - 1.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.26 - 7.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + (-3.22 - 5.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.47 + 2.55i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.02710477472978976107128267098, −10.87373249170389328167896898552, −9.237690122916806252069556134291, −8.714424180611981251679681601558, −8.196524394157723230039959240551, −6.59009233623364127776059251702, −5.61861132417287734365422243295, −4.73572564315703452396282763415, −2.50129024771432714440626709300, −1.52413947575859818087780277378, 2.20249323023959182286148971080, 3.67627620266510496567358777242, 4.56230258661595443503977275590, 6.06239196991667572067240759968, 7.12057584585226747175921377484, 8.124976283543907282423046150637, 9.372438280083538863619443095077, 10.29576072958740262969835692185, 10.87642818387585725501542516027, 11.37201835923871476052375034425

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