Properties

Degree $2$
Conductor $1152$
Sign $0.996 - 0.0805i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.00 − 1.00i)5-s + 10.0·7-s + (2.26 + 2.26i)11-s + (6.88 + 6.88i)13-s + 22.3·17-s + (16.8 − 16.8i)19-s − 33.2·23-s + 22.9i·25-s + (−24.6 − 24.6i)29-s + 41.3i·31-s + (10.1 − 10.1i)35-s + (6.60 − 6.60i)37-s − 47.1i·41-s + (48.8 + 48.8i)43-s + 45.6i·47-s + ⋯
L(s)  = 1  + (0.201 − 0.201i)5-s + 1.43·7-s + (0.205 + 0.205i)11-s + (0.529 + 0.529i)13-s + 1.31·17-s + (0.889 − 0.889i)19-s − 1.44·23-s + 0.918i·25-s + (−0.849 − 0.849i)29-s + 1.33i·31-s + (0.288 − 0.288i)35-s + (0.178 − 0.178i)37-s − 1.14i·41-s + (1.13 + 1.13i)43-s + 0.970i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0805i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.996 - 0.0805i$
Motivic weight: \(2\)
Character: $\chi_{1152} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.996 - 0.0805i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.647182855\)
\(L(\frac12)\) \(\approx\) \(2.647182855\)
\(L(2)\) 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 \( 1 + (-1.00 + 1.00i)T - 25iT^{2} \)
7 \( 1 - 10.0T + 49T^{2} \)
11 \( 1 + (-2.26 - 2.26i)T + 121iT^{2} \)
13 \( 1 + (-6.88 - 6.88i)T + 169iT^{2} \)
17 \( 1 - 22.3T + 289T^{2} \)
19 \( 1 + (-16.8 + 16.8i)T - 361iT^{2} \)
23 \( 1 + 33.2T + 529T^{2} \)
29 \( 1 + (24.6 + 24.6i)T + 841iT^{2} \)
31 \( 1 - 41.3iT - 961T^{2} \)
37 \( 1 + (-6.60 + 6.60i)T - 1.36e3iT^{2} \)
41 \( 1 + 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (-48.8 - 48.8i)T + 1.84e3iT^{2} \)
47 \( 1 - 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (-25.1 + 25.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-6.23 - 6.23i)T + 3.48e3iT^{2} \)
61 \( 1 + (35.9 + 35.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (10.2 - 10.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 11.9T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 + 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−9.494840581169859417375828635011, −8.867720528988750981275406195627, −7.81729164573521748436031746420, −7.43799433190426871564280858257, −6.10109470764836545521847649089, −5.31572049255421403205646587972, −4.50747438096748805461078783147, −3.47085452020797209439342518220, −1.99569463472750320179955785640, −1.12631934102959676460397941700, 1.03007674172492893200744338765, 2.05825404464816112068125282136, 3.43208146055349208684122423699, 4.33692734745026519639126498511, 5.58874732876416907276256480034, 5.88464731876803842207129363066, 7.38611980407161947761274288245, 7.949939650795687110263067242524, 8.567973607606187297865196793050, 9.771830106198472039355948344348

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