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Properties

Label 2-96-12.11-c1-0-2
Degree $2$
Conductor $96$
Sign $0.985 + 0.169i$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − i)3-s + 2.82i·5-s − 2i·7-s + (1.00 − 2.82i)9-s − 2.82·11-s − 2·13-s + (2.82 + 4.00i)15-s + 6i·19-s + (−2 − 2.82i)21-s − 5.65·23-s − 3.00·25-s + (−1.41 − 5.00i)27-s − 2.82i·29-s − 2i·31-s + (−4.00 + 2.82i)33-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + 1.26i·5-s − 0.755i·7-s + (0.333 − 0.942i)9-s − 0.852·11-s − 0.554·13-s + (0.730 + 1.03i)15-s + 1.37i·19-s + (−0.436 − 0.617i)21-s − 1.17·23-s − 0.600·25-s + (−0.272 − 0.962i)27-s − 0.525i·29-s − 0.359i·31-s + (−0.696 + 0.492i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18762 - 0.101142i\)
\(L(\frac12)\) \(\approx\) \(1.18762 - 0.101142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 8.48iT - 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + 16.9iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−14.08812374946186900952054595031, −13.11549341941892268370743631611, −11.92000599721075887174399465564, −10.52717864377721696717650463318, −9.803896087666515595975248140160, −8.011798526014807660749216778396, −7.37939068796191089832665455699, −6.16606299381823662093410340346, −3.85005515788955566299311392886, −2.45743383473735650058526571696, 2.52327980011448913469364768265, 4.48180512233011040460855361201, 5.44805557598716003682004686246, 7.62665210053119989042662015850, 8.719355882125173065570738118366, 9.344325113368977012383708582249, 10.59999566948605014169957353806, 12.11961282775681647891167740788, 12.99638929096153579145285892266, 13.94461742805014315394279729389

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