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Properties

Label 2-288-8.5-c7-0-14
Degree $2$
Conductor $288$
Sign $0.829 + 0.558i$
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  + 425. i·5-s − 1.66e3·7-s − 2.46e3i·11-s + 3.76e3i·13-s − 1.62e4·17-s + 4.86e3i·19-s − 1.08e5·23-s − 1.02e5·25-s − 8.90e4i·29-s + 6.94e4·31-s − 7.07e5i·35-s + 4.18e5i·37-s − 2.74e5·41-s − 4.62e5i·43-s + 1.53e5·47-s + ⋯
L(s)  = 1  + 1.52i·5-s − 1.83·7-s − 0.558i·11-s + 0.475i·13-s − 0.801·17-s + 0.162i·19-s − 1.86·23-s − 1.31·25-s − 0.678i·29-s + 0.418·31-s − 2.79i·35-s + 1.35i·37-s − 0.621·41-s − 0.887i·43-s + 0.215·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.829 + 0.558i$
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: \((2,\ 288,\ (\ :7/2),\ 0.829 + 0.558i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5223380010\)
\(L(\frac12)\) \(\approx\) \(0.5223380010\)
\(L(\frac{9}{2})\) 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 \( 1 - 425. iT - 7.81e4T^{2} \)
7 \( 1 + 1.66e3T + 8.23e5T^{2} \)
11 \( 1 + 2.46e3iT - 1.94e7T^{2} \)
13 \( 1 - 3.76e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.62e4T + 4.10e8T^{2} \)
19 \( 1 - 4.86e3iT - 8.93e8T^{2} \)
23 \( 1 + 1.08e5T + 3.40e9T^{2} \)
29 \( 1 + 8.90e4iT - 1.72e10T^{2} \)
31 \( 1 - 6.94e4T + 2.75e10T^{2} \)
37 \( 1 - 4.18e5iT - 9.49e10T^{2} \)
41 \( 1 + 2.74e5T + 1.94e11T^{2} \)
43 \( 1 + 4.62e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.53e5T + 5.06e11T^{2} \)
53 \( 1 - 1.81e5iT - 1.17e12T^{2} \)
59 \( 1 + 6.47e5iT - 2.48e12T^{2} \)
61 \( 1 - 2.26e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.31e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.77e6T + 9.09e12T^{2} \)
73 \( 1 + 4.45e6T + 1.10e13T^{2} \)
79 \( 1 - 3.14e6T + 1.92e13T^{2} \)
83 \( 1 + 3.33e6iT - 2.71e13T^{2} \)
89 \( 1 - 5.00e6T + 4.42e13T^{2} \)
97 \( 1 - 8.06e6T + 8.07e13T^{2} \)
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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

−10.23404077780998558305216479843, −9.916749555619468727491715136867, −8.664321249842791906818914832651, −7.31416432545538799434702121862, −6.44043495127198970164518930727, −6.05296054723286454358364717902, −4.00547171925748620636741498260, −3.17237087230401423633176681643, −2.27862763233520047667091599675, −0.18484947167765182824508796086, 0.61610582270047099924719395989, 2.11204258346122917082341898164, 3.55641189709587319022402025488, 4.57565213077528529849079430972, 5.73839857375775704710031255976, 6.64006671757658033271087138152, 7.905787394130074511279923271888, 8.984352206301862356409780413900, 9.573144983427950876356124415998, 10.41415155252353737024771687000

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