Properties

Degree $2$
Conductor $576$
Sign $0.0277 - 0.999i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 − 1.69i)5-s + 5.74·7-s + (−5.59 + 5.59i)11-s + (−13.5 + 13.5i)13-s − 19.7·17-s + (21.6 + 21.6i)19-s + 24.9·23-s − 19.2i·25-s + (−1.50 + 1.50i)29-s − 2.20i·31-s + (−9.75 − 9.75i)35-s + (27.6 + 27.6i)37-s + 51.3i·41-s + (−21.4 + 21.4i)43-s + 76.5i·47-s + ⋯
L(s)  = 1  + (−0.339 − 0.339i)5-s + 0.820·7-s + (−0.508 + 0.508i)11-s + (−1.04 + 1.04i)13-s − 1.15·17-s + (1.14 + 1.14i)19-s + 1.08·23-s − 0.768i·25-s + (−0.0519 + 0.0519i)29-s − 0.0709i·31-s + (−0.278 − 0.278i)35-s + (0.748 + 0.748i)37-s + 1.25i·41-s + (−0.498 + 0.498i)43-s + 1.62i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0277 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0277 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.0277 - 0.999i$
Motivic weight: \(2\)
Character: $\chi_{576} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.0277 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.249501660\)
\(L(\frac12)\) \(\approx\) \(1.249501660\)
\(L(2)\) 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 \( 1 + (1.69 + 1.69i)T + 25iT^{2} \)
7 \( 1 - 5.74T + 49T^{2} \)
11 \( 1 + (5.59 - 5.59i)T - 121iT^{2} \)
13 \( 1 + (13.5 - 13.5i)T - 169iT^{2} \)
17 \( 1 + 19.7T + 289T^{2} \)
19 \( 1 + (-21.6 - 21.6i)T + 361iT^{2} \)
23 \( 1 - 24.9T + 529T^{2} \)
29 \( 1 + (1.50 - 1.50i)T - 841iT^{2} \)
31 \( 1 + 2.20iT - 961T^{2} \)
37 \( 1 + (-27.6 - 27.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 51.3iT - 1.68e3T^{2} \)
43 \( 1 + (21.4 - 21.4i)T - 1.84e3iT^{2} \)
47 \( 1 - 76.5iT - 2.20e3T^{2} \)
53 \( 1 + (-56.5 - 56.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (48.0 - 48.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (51.5 - 51.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (63.4 + 63.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 43.4T + 5.04e3T^{2} \)
73 \( 1 + 73.9iT - 5.32e3T^{2} \)
79 \( 1 + 4.12iT - 6.24e3T^{2} \)
83 \( 1 + (-38.4 - 38.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 52.9iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−10.81192876481013941183597601210, −9.773441533161114908858417247934, −9.022437062494272609986679310110, −7.942800866320623276924836107867, −7.37457093920595181985041226462, −6.21048107223486360894834822568, −4.77745110293056811790566609737, −4.51340248564347196959885807434, −2.77295593512349460723312794354, −1.47882197198437009980606858990, 0.48340419852237540106546152177, 2.35403545958006047835977103739, 3.39267639491642012953389169449, 4.88936593301569573222621473236, 5.40120281624879487233116799475, 6.97707289787636929058634757280, 7.52344391231144563775666823942, 8.498543991980779189239244139270, 9.369799890737295433903200884279, 10.51416288541585355583218684588

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