Properties

Degree $2$
Conductor $1152$
Sign $0.543 + 0.839i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.543 + 0.839i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.543 + 0.839i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.193400906\)
\(L(\frac12)\) \(\approx\) \(2.193400906\)
\(L(2)\) 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

−9.331746144605593969033369023687, −8.601840518753123607136941454003, −8.046255909940862640424679945331, −6.75706607654223005736909353589, −6.27752530602615884705202794511, −5.16307136668455526112161770787, −4.30033152939215935474702379367, −3.12843049326683609110562185820, −2.04755688364247972250152027556, −0.68900672991576514952107912250, 1.38227493158715346966553792292, 2.07499097388912493587729900312, 3.78158201953847751815043127377, 4.48914750763648422179994551495, 5.43645166219747105564739660273, 6.48774960327278909418254404281, 7.09549631741662557130278893272, 8.414316259530539994409591129927, 8.775245263710057772618967383965, 9.595416320586724267539869890724

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