Properties

Degree $2$
Conductor $1152$
Sign $-0.744 + 0.668i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.49 − 6.49i)5-s − 3.94·7-s + (−4.31 − 4.31i)11-s + (−4.06 − 4.06i)13-s + 14.5·17-s + (4.94 − 4.94i)19-s − 43.6·23-s − 59.3i·25-s + (25.0 + 25.0i)29-s − 32.5i·31-s + (−25.6 + 25.6i)35-s + (−4.14 + 4.14i)37-s − 55.3i·41-s + (−16.1 − 16.1i)43-s − 7.92i·47-s + ⋯
L(s)  = 1  + (1.29 − 1.29i)5-s − 0.563·7-s + (−0.391 − 0.391i)11-s + (−0.312 − 0.312i)13-s + 0.856·17-s + (0.260 − 0.260i)19-s − 1.89·23-s − 2.37i·25-s + (0.865 + 0.865i)29-s − 1.04i·31-s + (−0.731 + 0.731i)35-s + (−0.111 + 0.111i)37-s − 1.34i·41-s + (−0.374 − 0.374i)43-s − 0.168i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.668i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.744 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.744 + 0.668i$
Motivic weight: \(2\)
Character: $\chi_{1152} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.744 + 0.668i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.604526709\)
\(L(\frac12)\) \(\approx\) \(1.604526709\)
\(L(2)\) 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 \( 1 + (-6.49 + 6.49i)T - 25iT^{2} \)
7 \( 1 + 3.94T + 49T^{2} \)
11 \( 1 + (4.31 + 4.31i)T + 121iT^{2} \)
13 \( 1 + (4.06 + 4.06i)T + 169iT^{2} \)
17 \( 1 - 14.5T + 289T^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 361iT^{2} \)
23 \( 1 + 43.6T + 529T^{2} \)
29 \( 1 + (-25.0 - 25.0i)T + 841iT^{2} \)
31 \( 1 + 32.5iT - 961T^{2} \)
37 \( 1 + (4.14 - 4.14i)T - 1.36e3iT^{2} \)
41 \( 1 + 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (16.1 + 16.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (31.5 - 31.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-49.7 - 49.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (44.4 + 44.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (1.64 - 1.64i)T - 4.48e3iT^{2} \)
71 \( 1 - 24.1T + 5.04e3T^{2} \)
73 \( 1 - 10.7iT - 5.32e3T^{2} \)
79 \( 1 - 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (42.0 - 42.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−9.446183443442738512404298908268, −8.535775457509294502406389321192, −7.86124036086795563913808051211, −6.57152391486190795607019704313, −5.68781405869014937786478343087, −5.28134938538783841023145382499, −4.11740819990294260698105849409, −2.79260313663932999445440940287, −1.68170871816682751610892755765, −0.44622897060741299350365834320, 1.71375390394043296079190125245, 2.65935392321590522364223710964, 3.48744960266391095485553086463, 4.90633636592556081233246308145, 6.01114820893045290327678402305, 6.39037865761411752508348977372, 7.32121686600107100347934209644, 8.172050559676732350012484327570, 9.538077171199134787699108088282, 9.989123204920371852444404421434

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