Properties

Label 2-288-8.5-c7-0-17
Degree $2$
Conductor $288$
Sign $0.985 + 0.167i$
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  − 338. i·5-s + 438.·7-s + 1.96e3i·11-s − 2.21e3i·13-s + 1.21e4·17-s + 3.28e4i·19-s + 1.96e4·23-s − 3.64e4·25-s + 1.60e5i·29-s + 2.29e5·31-s − 1.48e5i·35-s + 4.96e5i·37-s − 5.99e5·41-s + 8.83e4i·43-s + 8.20e5·47-s + ⋯
L(s)  = 1  − 1.21i·5-s + 0.483·7-s + 0.445i·11-s − 0.279i·13-s + 0.598·17-s + 1.09i·19-s + 0.335·23-s − 0.466·25-s + 1.22i·29-s + 1.38·31-s − 0.585i·35-s + 1.61i·37-s − 1.35·41-s + 0.169i·43-s + 1.15·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.985 + 0.167i$
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: \((2,\ 288,\ (\ :7/2),\ 0.985 + 0.167i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.370923539\)
\(L(\frac12)\) \(\approx\) \(2.370923539\)
\(L(\frac{9}{2})\) 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 \( 1 + 338. iT - 7.81e4T^{2} \)
7 \( 1 - 438.T + 8.23e5T^{2} \)
11 \( 1 - 1.96e3iT - 1.94e7T^{2} \)
13 \( 1 + 2.21e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.21e4T + 4.10e8T^{2} \)
19 \( 1 - 3.28e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.96e4T + 3.40e9T^{2} \)
29 \( 1 - 1.60e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.29e5T + 2.75e10T^{2} \)
37 \( 1 - 4.96e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.99e5T + 1.94e11T^{2} \)
43 \( 1 - 8.83e4iT - 2.71e11T^{2} \)
47 \( 1 - 8.20e5T + 5.06e11T^{2} \)
53 \( 1 + 1.53e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.82e6iT - 2.48e12T^{2} \)
61 \( 1 - 4.84e5iT - 3.14e12T^{2} \)
67 \( 1 + 7.98e4iT - 6.06e12T^{2} \)
71 \( 1 - 1.27e6T + 9.09e12T^{2} \)
73 \( 1 - 3.70e6T + 1.10e13T^{2} \)
79 \( 1 - 2.55e6T + 1.92e13T^{2} \)
83 \( 1 + 1.53e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.99e6T + 4.42e13T^{2} \)
97 \( 1 + 2.89e4T + 8.07e13T^{2} \)
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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

−10.42481691119632914629727034228, −9.624049666455980270849210030079, −8.485466475044970661812401597189, −7.977662555913629725375612881634, −6.62746481966401811168587596921, −5.29094975038688718407509724635, −4.71300457931473001014492138410, −3.37186127622600639590274552563, −1.72827778681345480184168019995, −0.871403355405954173297565186615, 0.70824623523083055574386937919, 2.27285804578250109880161395284, 3.20498114826299436883931411724, 4.47053645442175440718556339731, 5.76832391031901286582002096976, 6.76636235205505241779469997919, 7.59688339660112830535011056259, 8.671291884078554147103855074789, 9.794547540289659369184348650118, 10.75476955558294078218536297553

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