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Properties

Label 2-96-8.5-c3-0-5
Degree $2$
Conductor $96$
Sign $-0.896 + 0.442i$
Analytic cond. $5.66418$
Root an. cond. $2.37995$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 9.15i·5-s − 27.4·7-s − 9·9-s + 20.5i·11-s − 32.0i·13-s − 27.4·15-s − 111.·17-s − 129. i·19-s + 82.2i·21-s − 9.16·23-s + 41.1·25-s + 27i·27-s − 41.0i·29-s + 187.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.818i·5-s − 1.48·7-s − 0.333·9-s + 0.562i·11-s − 0.683i·13-s − 0.472·15-s − 1.59·17-s − 1.56i·19-s + 0.854i·21-s − 0.0830·23-s + 0.329·25-s + 0.192i·27-s − 0.262i·29-s + 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(5.66418\)
Root analytic conductor: \(2.37995\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :3/2),\ -0.896 + 0.442i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.175967 - 0.753364i\)
\(L(\frac12)\) \(\approx\) \(0.175967 - 0.753364i\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
good5 \( 1 + 9.15iT - 125T^{2} \)
7 \( 1 + 27.4T + 343T^{2} \)
11 \( 1 - 20.5iT - 1.33e3T^{2} \)
13 \( 1 + 32.0iT - 2.19e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 + 129. iT - 6.85e3T^{2} \)
23 \( 1 + 9.16T + 1.21e4T^{2} \)
29 \( 1 + 41.0iT - 2.43e4T^{2} \)
31 \( 1 - 187.T + 2.97e4T^{2} \)
37 \( 1 + 114. iT - 5.06e4T^{2} \)
41 \( 1 - 282.T + 6.89e4T^{2} \)
43 \( 1 + 89.3iT - 7.95e4T^{2} \)
47 \( 1 - 54.6T + 1.03e5T^{2} \)
53 \( 1 - 726. iT - 1.48e5T^{2} \)
59 \( 1 + 216. iT - 2.05e5T^{2} \)
61 \( 1 + 754. iT - 2.26e5T^{2} \)
67 \( 1 + 379. iT - 3.00e5T^{2} \)
71 \( 1 + 302.T + 3.57e5T^{2} \)
73 \( 1 + 504.T + 3.89e5T^{2} \)
79 \( 1 + 301.T + 4.93e5T^{2} \)
83 \( 1 + 599. iT - 5.71e5T^{2} \)
89 \( 1 + 277.T + 7.04e5T^{2} \)
97 \( 1 + 765.T + 9.12e5T^{2} \)
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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

−12.97460233467944982781914892342, −12.35895739174467119353599773459, −10.92951141871600687948614241603, −9.531020535778768993725954081761, −8.725230393358586954673344103640, −7.18925671613010752966546073129, −6.20005229814512931253778175358, −4.59911670497521363428999385902, −2.68575984375190588330567942750, −0.42649129458420919847436324333, 2.84130777782109793203754866122, 4.06619187001677496390527624704, 6.05078342732781860751208086385, 6.85526711233969870412937232531, 8.622449718738678300323171465422, 9.742735484149604615352361279147, 10.57152655876415591547269649038, 11.66360683247368478331866392769, 12.97885982329157132518997640069, 13.97688373945185344773868411510

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