Properties

Degree $2$
Conductor $1152$
Sign $0.999 + 0.00671i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.78 − 4.78i)5-s − 10.3·7-s + (−0.526 + 0.526i)11-s + (−17.2 + 17.2i)13-s − 4.71·17-s + (2.53 + 2.53i)19-s + 12.5·23-s + 20.8i·25-s + (−2.19 + 2.19i)29-s − 28.0i·31-s + (49.4 + 49.4i)35-s + (32.1 + 32.1i)37-s − 23.1i·41-s + (−4.79 + 4.79i)43-s − 39.0i·47-s + ⋯
L(s)  = 1  + (−0.957 − 0.957i)5-s − 1.47·7-s + (−0.0478 + 0.0478i)11-s + (−1.32 + 1.32i)13-s − 0.277·17-s + (0.133 + 0.133i)19-s + 0.547·23-s + 0.834i·25-s + (−0.0757 + 0.0757i)29-s − 0.904i·31-s + (1.41 + 1.41i)35-s + (0.867 + 0.867i)37-s − 0.563i·41-s + (−0.111 + 0.111i)43-s − 0.829i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.999 + 0.00671i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.999 + 0.00671i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7225329894\)
\(L(\frac12)\) \(\approx\) \(0.7225329894\)
\(L(2)\) 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 \( 1 + (4.78 + 4.78i)T + 25iT^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + (0.526 - 0.526i)T - 121iT^{2} \)
13 \( 1 + (17.2 - 17.2i)T - 169iT^{2} \)
17 \( 1 + 4.71T + 289T^{2} \)
19 \( 1 + (-2.53 - 2.53i)T + 361iT^{2} \)
23 \( 1 - 12.5T + 529T^{2} \)
29 \( 1 + (2.19 - 2.19i)T - 841iT^{2} \)
31 \( 1 + 28.0iT - 961T^{2} \)
37 \( 1 + (-32.1 - 32.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 23.1iT - 1.68e3T^{2} \)
43 \( 1 + (4.79 - 4.79i)T - 1.84e3iT^{2} \)
47 \( 1 + 39.0iT - 2.20e3T^{2} \)
53 \( 1 + (27.9 + 27.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-79.8 + 79.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (-36.7 + 36.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-10.9 - 10.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 52.6T + 5.04e3T^{2} \)
73 \( 1 - 67.8iT - 5.32e3T^{2} \)
79 \( 1 + 56.4iT - 6.24e3T^{2} \)
83 \( 1 + (58.3 + 58.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 - 60.9T + 9.40e3T^{2} \)
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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

−9.513195352125931145582462433164, −8.935850101075332083409327861116, −7.965148124322713339475663205205, −7.07516310233099046496856374769, −6.44169470118386819940595994946, −5.15533074711540325000937991897, −4.35957761836568765736502906313, −3.52507559968836338808885153539, −2.27986404076872402641575823610, −0.54292822189514017021420904768, 0.43130828294543114500710861957, 2.78915727400149060813222056347, 3.08977708324495851708654646073, 4.20470712469692099644450984888, 5.42845865419309550388750751169, 6.42012430524539809875510484842, 7.23383721315368271444815770320, 7.65726354718871729799489191212, 8.833391264601046049648730710372, 9.790753913266241288793745958826

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