Properties

Label 2-96-4.3-c2-0-3
Degree $2$
Conductor $96$
Sign $-0.707 + 0.707i$
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  − 1.73i·3-s − 8.92·5-s − 10.9i·7-s − 2.99·9-s + 1.07i·11-s − 3.85·13-s + 15.4i·15-s + 7.85·17-s − 17.0i·19-s − 18.9·21-s − 8i·23-s + 54.7·25-s + 5.19i·27-s − 3.07·29-s + 30.6i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.78·5-s − 1.56i·7-s − 0.333·9-s + 0.0974i·11-s − 0.296·13-s + 1.03i·15-s + 0.462·17-s − 0.898i·19-s − 0.901·21-s − 0.347i·23-s + 2.18·25-s + 0.192i·27-s − 0.105·29-s + 0.988i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(2.61581\)
Root analytic conductor: \(1.61734\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.270529 - 0.653115i\)
\(L(\frac12)\) \(\approx\) \(0.270529 - 0.653115i\)
\(L(2)\) 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 \( 1 + 1.73iT \)
good5 \( 1 + 8.92T + 25T^{2} \)
7 \( 1 + 10.9iT - 49T^{2} \)
11 \( 1 - 1.07iT - 121T^{2} \)
13 \( 1 + 3.85T + 169T^{2} \)
17 \( 1 - 7.85T + 289T^{2} \)
19 \( 1 + 17.0iT - 361T^{2} \)
23 \( 1 + 8iT - 529T^{2} \)
29 \( 1 + 3.07T + 841T^{2} \)
31 \( 1 - 30.6iT - 961T^{2} \)
37 \( 1 - 45.7T + 1.36e3T^{2} \)
41 \( 1 + 35.8T + 1.68e3T^{2} \)
43 \( 1 + 74.6iT - 1.84e3T^{2} \)
47 \( 1 + 42.1iT - 2.20e3T^{2} \)
53 \( 1 - 12.9T + 2.80e3T^{2} \)
59 \( 1 + 44.2iT - 3.48e3T^{2} \)
61 \( 1 + 14T + 3.72e3T^{2} \)
67 \( 1 - 80.4iT - 4.48e3T^{2} \)
71 \( 1 + 123. iT - 5.04e3T^{2} \)
73 \( 1 + 85.4T + 5.32e3T^{2} \)
79 \( 1 - 55.2iT - 6.24e3T^{2} \)
83 \( 1 + 49.0iT - 6.88e3T^{2} \)
89 \( 1 + 105.T + 7.92e3T^{2} \)
97 \( 1 - 21.1T + 9.40e3T^{2} \)
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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

−13.26176141351711986281715509501, −12.20163027848233753574958005000, −11.31848360664252153439381015016, −10.35868428350265203865940931403, −8.542383396274933464542890600772, −7.47502093493993403547271559668, −6.96643920361716381249847993792, −4.64026790908089646820446311201, −3.47927522361454661187411105599, −0.55621185081534420143821240295, 3.08297745136434997818303319342, 4.44321180121934181129649572106, 5.85480142658184004903502532458, 7.69849353166098152506753740818, 8.512428773691668608905561372131, 9.688611660011099148945590703899, 11.25273628267354599161625219392, 11.85629303770758055565144121831, 12.69275416098322429696510957707, 14.63409366439559393652436584326

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