Properties

Label 2-288-8.5-c7-0-0
Degree $2$
Conductor $288$
Sign $-0.857 + 0.513i$
Analytic cond. $89.9668$
Root an. cond. $9.48508$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 324. i·5-s + 956.·7-s − 5.45e3i·11-s + 6.28e3i·13-s − 3.45e4·17-s − 1.45e4i·19-s − 2.46e4·23-s − 2.71e4·25-s + 1.71e5i·29-s − 1.11e5·31-s + 3.10e5i·35-s + 1.03e5i·37-s − 7.16e4·41-s − 3.28e5i·43-s + 1.19e5·47-s + ⋯
L(s)  = 1  + 1.16i·5-s + 1.05·7-s − 1.23i·11-s + 0.793i·13-s − 1.70·17-s − 0.488i·19-s − 0.422·23-s − 0.347·25-s + 1.30i·29-s − 0.673·31-s + 1.22i·35-s + 0.336i·37-s − 0.162·41-s − 0.629i·43-s + 0.167·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.513i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.857 + 0.513i$
Analytic conductor: \(89.9668\)
Root analytic conductor: \(9.48508\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :7/2),\ -0.857 + 0.513i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.03319267175\)
\(L(\frac12)\) \(\approx\) \(0.03319267175\)
\(L(\frac{9}{2})\) 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 \( 1 \)
good5 \( 1 - 324. iT - 7.81e4T^{2} \)
7 \( 1 - 956.T + 8.23e5T^{2} \)
11 \( 1 + 5.45e3iT - 1.94e7T^{2} \)
13 \( 1 - 6.28e3iT - 6.27e7T^{2} \)
17 \( 1 + 3.45e4T + 4.10e8T^{2} \)
19 \( 1 + 1.45e4iT - 8.93e8T^{2} \)
23 \( 1 + 2.46e4T + 3.40e9T^{2} \)
29 \( 1 - 1.71e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.11e5T + 2.75e10T^{2} \)
37 \( 1 - 1.03e5iT - 9.49e10T^{2} \)
41 \( 1 + 7.16e4T + 1.94e11T^{2} \)
43 \( 1 + 3.28e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.19e5T + 5.06e11T^{2} \)
53 \( 1 + 1.04e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.25e5iT - 2.48e12T^{2} \)
61 \( 1 + 1.55e6iT - 3.14e12T^{2} \)
67 \( 1 - 3.16e5iT - 6.06e12T^{2} \)
71 \( 1 - 5.38e5T + 9.09e12T^{2} \)
73 \( 1 + 2.68e6T + 1.10e13T^{2} \)
79 \( 1 + 8.22e6T + 1.92e13T^{2} \)
83 \( 1 - 5.89e6iT - 2.71e13T^{2} \)
89 \( 1 + 4.37e5T + 4.42e13T^{2} \)
97 \( 1 + 7.84e6T + 8.07e13T^{2} \)
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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

−11.15350406965528913200002403367, −10.54234957680665203228118903201, −9.077158143003600146507266343876, −8.362108303672253250773397974997, −7.11099124764168262966228790107, −6.44220810204391801722733248322, −5.14771507170683438146214607249, −3.96058232578594535571834582775, −2.74905175919918062335940748359, −1.66084392326879090507353123228, 0.00693295924804094371925221661, 1.35559526038004661783258325906, 2.27413555065958061017605826389, 4.25350040481324500070445983841, 4.76576137186453328773351532869, 5.86251957187990376368455940599, 7.31393585123260062583041186088, 8.184694959668858853792734069793, 8.966267380438181092691916257170, 9.971958285575521268507654097567

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