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Properties

Label 2-96-24.11-c3-0-9
Degree $2$
Conductor $96$
Sign $-0.190 + 0.981i$
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  + (4.37 − 2.80i)3-s − 13.1·5-s − 26.9i·7-s + (11.2 − 24.5i)9-s − 21.9i·11-s + 10.0i·13-s + (−57.5 + 36.9i)15-s − 6.09i·17-s + 40.1·19-s + (−75.6 − 117. i)21-s + 9.80·23-s + 48.4·25-s + (−19.8 − 138. i)27-s + 164.·29-s − 47.0i·31-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s − 1.17·5-s − 1.45i·7-s + (0.416 − 0.909i)9-s − 0.602i·11-s + 0.214i·13-s + (−0.991 + 0.636i)15-s − 0.0869i·17-s + 0.485·19-s + (−0.786 − 1.22i)21-s + 0.0889·23-s + 0.387·25-s + (−0.141 − 0.989i)27-s + 1.05·29-s − 0.272i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(5.66418\)
Root analytic conductor: \(2.37995\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :3/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.968906 - 1.17453i\)
\(L(\frac12)\) \(\approx\) \(0.968906 - 1.17453i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.37 + 2.80i)T \)
good5 \( 1 + 13.1T + 125T^{2} \)
7 \( 1 + 26.9iT - 343T^{2} \)
11 \( 1 + 21.9iT - 1.33e3T^{2} \)
13 \( 1 - 10.0iT - 2.19e3T^{2} \)
17 \( 1 + 6.09iT - 4.91e3T^{2} \)
19 \( 1 - 40.1T + 6.85e3T^{2} \)
23 \( 1 - 9.80T + 1.21e4T^{2} \)
29 \( 1 - 164.T + 2.43e4T^{2} \)
31 \( 1 + 47.0iT - 2.97e4T^{2} \)
37 \( 1 - 205. iT - 5.06e4T^{2} \)
41 \( 1 - 419. iT - 6.89e4T^{2} \)
43 \( 1 + 205.T + 7.95e4T^{2} \)
47 \( 1 - 566.T + 1.03e5T^{2} \)
53 \( 1 - 342.T + 1.48e5T^{2} \)
59 \( 1 + 3.70iT - 2.05e5T^{2} \)
61 \( 1 + 717. iT - 2.26e5T^{2} \)
67 \( 1 - 238.T + 3.00e5T^{2} \)
71 \( 1 + 517.T + 3.57e5T^{2} \)
73 \( 1 - 984.T + 3.89e5T^{2} \)
79 \( 1 - 329. iT - 4.93e5T^{2} \)
83 \( 1 + 625. iT - 5.71e5T^{2} \)
89 \( 1 + 238. iT - 7.04e5T^{2} \)
97 \( 1 + 1.00e3T + 9.12e5T^{2} \)
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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

−13.42847605618750152860698858920, −12.15878230663123928600149919461, −11.16277819927138299538904770878, −9.876756494040426168268779386326, −8.404114026334948359352032495232, −7.63562855736328386374447744952, −6.70582404642314802210382651244, −4.31041066056593233778305906958, −3.27599412203413946463570699143, −0.849022138233924202297986495240, 2.53347164920422888289101515354, 3.94549542907257371470297711489, 5.32733759009802854342328596374, 7.31098011341241416621144944556, 8.388866150550573421216096204584, 9.172302648520127061921639298192, 10.47542910010543491293670930663, 11.82411275763424909429102597546, 12.52116923779167874000775381365, 13.98124372004876670468762238149

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