Properties

Label 2-96-3.2-c2-0-5
Degree $2$
Conductor $96$
Sign $0.816 + 0.577i$
Analytic cond. $2.61581$
Root an. cond. $1.61734$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 2.44i)3-s + 2.82i·5-s + 10.3·7-s + (−2.99 − 8.48i)9-s − 14.6i·11-s − 6·13-s + (6.92 + 4.89i)15-s + 22.6i·17-s − 10.3·19-s + (18 − 25.4i)21-s + 29.3i·23-s + 17·25-s + (−25.9 − 7.34i)27-s + 31.1i·29-s − 31.1·31-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + 0.565i·5-s + 1.48·7-s + (−0.333 − 0.942i)9-s − 1.33i·11-s − 0.461·13-s + (0.461 + 0.326i)15-s + 1.33i·17-s − 0.546·19-s + (0.857 − 1.21i)21-s + 1.27i·23-s + 0.680·25-s + (−0.962 − 0.272i)27-s + 1.07i·29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(2.61581\)
Root analytic conductor: \(1.61734\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59317 - 0.506370i\)
\(L(\frac12)\) \(\approx\) \(1.59317 - 0.506370i\)
\(L(2)\) 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 \( 1 + (-1.73 + 2.44i)T \)
good5 \( 1 - 2.82iT - 25T^{2} \)
7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 + 6T + 169T^{2} \)
17 \( 1 - 22.6iT - 289T^{2} \)
19 \( 1 + 10.3T + 361T^{2} \)
23 \( 1 - 29.3iT - 529T^{2} \)
29 \( 1 - 31.1iT - 841T^{2} \)
31 \( 1 + 31.1T + 961T^{2} \)
37 \( 1 + 38T + 1.36e3T^{2} \)
41 \( 1 + 5.65iT - 1.68e3T^{2} \)
43 \( 1 + 10.3T + 1.84e3T^{2} \)
47 \( 1 + 58.7iT - 2.20e3T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + 14.6iT - 3.48e3T^{2} \)
61 \( 1 + 22T + 3.72e3T^{2} \)
67 \( 1 - 114.T + 4.48e3T^{2} \)
71 \( 1 + 29.3iT - 5.04e3T^{2} \)
73 \( 1 + 30T + 5.32e3T^{2} \)
79 \( 1 + 31.1T + 6.24e3T^{2} \)
83 \( 1 - 73.4iT - 6.88e3T^{2} \)
89 \( 1 + 5.65iT - 7.92e3T^{2} \)
97 \( 1 - 90T + 9.40e3T^{2} \)
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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

−13.81776469499798066423443491148, −12.64872172252632554399853939551, −11.44611411471106840571153656298, −10.66839428990328392981943756966, −8.812947937059081909045973510623, −8.094474269897240955774009407728, −6.95910415703906066547271563213, −5.53115874669343422208814807699, −3.49732992569682746460584074973, −1.73105121822732431778926147941, 2.22366461339349790618826363007, 4.49778979338763824838428107830, 5.00158943917870760443133334421, 7.35533197862825724121992851566, 8.409685389483895331322001040769, 9.398825701952516204538508999135, 10.50835739070008868032694979114, 11.63029659353497320752910082357, 12.77274894796780927506349148343, 14.20835056765261540348153957951

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