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Properties

Label 2-96-8.3-c2-0-2
Degree $2$
Conductor $96$
Sign $0.183 + 0.983i$
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  − 1.73·3-s − 2.87i·5-s − 10.7i·7-s + 2.99·9-s + 8·11-s − 15.7i·13-s + 4.98i·15-s − 15.8·17-s − 1.07·19-s + 18.6i·21-s + 21.4i·23-s + 16.7·25-s − 5.19·27-s + 40.0i·29-s − 9.20i·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.575i·5-s − 1.53i·7-s + 0.333·9-s + 0.727·11-s − 1.20i·13-s + 0.332i·15-s − 0.932·17-s − 0.0564·19-s + 0.886i·21-s + 0.934i·23-s + 0.668·25-s − 0.192·27-s + 1.38i·29-s − 0.296i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.183 + 0.983i$
Analytic conductor: \(2.61581\)
Root analytic conductor: \(1.61734\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1),\ 0.183 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.800304 - 0.665070i\)
\(L(\frac12)\) \(\approx\) \(0.800304 - 0.665070i\)
\(L(2)\) 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 \( 1 + 1.73T \)
good5 \( 1 + 2.87iT - 25T^{2} \)
7 \( 1 + 10.7iT - 49T^{2} \)
11 \( 1 - 8T + 121T^{2} \)
13 \( 1 + 15.7iT - 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
19 \( 1 + 1.07T + 361T^{2} \)
23 \( 1 - 21.4iT - 529T^{2} \)
29 \( 1 - 40.0iT - 841T^{2} \)
31 \( 1 + 9.20iT - 961T^{2} \)
37 \( 1 + 9.97iT - 1.36e3T^{2} \)
41 \( 1 - 51.5T + 1.68e3T^{2} \)
43 \( 1 - 12.7T + 1.84e3T^{2} \)
47 \( 1 + 1.54iT - 2.20e3T^{2} \)
53 \( 1 + 28.5iT - 2.80e3T^{2} \)
59 \( 1 - 11.2T + 3.48e3T^{2} \)
61 \( 1 + 1.54iT - 3.72e3T^{2} \)
67 \( 1 - 43.2T + 4.48e3T^{2} \)
71 \( 1 + 84.4iT - 5.04e3T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 - 73.6iT - 6.24e3T^{2} \)
83 \( 1 + 12.2T + 6.88e3T^{2} \)
89 \( 1 - 33.1T + 7.92e3T^{2} \)
97 \( 1 + 69.1T + 9.40e3T^{2} \)
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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

−13.30897666460557281962270348330, −12.58767972774858125623673456406, −11.19017724179972350484322905527, −10.46185995999847872357331242159, −9.193307244819598524491510999011, −7.72527875843249215604856316257, −6.64185458853050468309583150079, −5.10212536718697014132861994491, −3.83767617505855486641527890487, −0.929413234716795791901505677024, 2.35503825140727610443945900475, 4.44289649370149009860206090090, 6.02012144720189482322314123805, 6.81294755735736500508846832778, 8.629283470338112893205699526480, 9.525964134755084538091811670113, 11.02653094609828288793054204242, 11.76582591312203443895420103541, 12.63963281812155654800069785944, 14.09573306024793519898269688668

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