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Properties

Label 2-96-96.5-c4-0-0
Degree $2$
Conductor $96$
Sign $-0.832 + 0.554i$
Analytic cond. $9.92351$
Root an. cond. $3.15016$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.95 + 0.622i)2-s + (8.32 + 3.41i)3-s + (15.2 − 4.91i)4-s + (−16.1 + 38.9i)5-s + (−35.0 − 8.32i)6-s + (−56.6 − 56.6i)7-s + (−57.0 + 28.9i)8-s + (57.6 + 56.9i)9-s + (39.4 − 163. i)10-s + (−142. − 58.9i)11-s + (143. + 11.1i)12-s + (37.2 − 15.4i)13-s + (259. + 188. i)14-s + (−267. + 269. i)15-s + (207. − 149. i)16-s + 143.·17-s + ⋯
L(s)  = 1  + (−0.987 + 0.155i)2-s + (0.925 + 0.379i)3-s + (0.951 − 0.307i)4-s + (−0.645 + 1.55i)5-s + (−0.972 − 0.231i)6-s + (−1.15 − 1.15i)7-s + (−0.892 + 0.451i)8-s + (0.711 + 0.702i)9-s + (0.394 − 1.63i)10-s + (−1.17 − 0.486i)11-s + (0.997 + 0.0771i)12-s + (0.220 − 0.0911i)13-s + (1.32 + 0.962i)14-s + (−1.18 + 1.19i)15-s + (0.811 − 0.584i)16-s + 0.498·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(9.92351\)
Root analytic conductor: \(3.15016\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :2),\ -0.832 + 0.554i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0374899 - 0.123872i\)
\(L(\frac12)\) \(\approx\) \(0.0374899 - 0.123872i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.95 - 0.622i)T \)
3 \( 1 + (-8.32 - 3.41i)T \)
good5 \( 1 + (16.1 - 38.9i)T + (-441. - 441. i)T^{2} \)
7 \( 1 + (56.6 + 56.6i)T + 2.40e3iT^{2} \)
11 \( 1 + (142. + 58.9i)T + (1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 + (-37.2 + 15.4i)T + (2.01e4 - 2.01e4i)T^{2} \)
17 \( 1 - 143.T + 8.35e4T^{2} \)
19 \( 1 + (-9.97 + 4.13i)T + (9.21e4 - 9.21e4i)T^{2} \)
23 \( 1 + (648. + 648. i)T + 2.79e5iT^{2} \)
29 \( 1 + (241. - 100. i)T + (5.00e5 - 5.00e5i)T^{2} \)
31 \( 1 + 1.27e3T + 9.23e5T^{2} \)
37 \( 1 + (-612. - 253. i)T + (1.32e6 + 1.32e6i)T^{2} \)
41 \( 1 + (-812. - 812. i)T + 2.82e6iT^{2} \)
43 \( 1 + (813. - 1.96e3i)T + (-2.41e6 - 2.41e6i)T^{2} \)
47 \( 1 + 3.38e3T + 4.87e6T^{2} \)
53 \( 1 + (-90.6 - 37.5i)T + (5.57e6 + 5.57e6i)T^{2} \)
59 \( 1 + (-428. + 1.03e3i)T + (-8.56e6 - 8.56e6i)T^{2} \)
61 \( 1 + (-997. - 2.40e3i)T + (-9.79e6 + 9.79e6i)T^{2} \)
67 \( 1 + (951. + 2.29e3i)T + (-1.42e7 + 1.42e7i)T^{2} \)
71 \( 1 + (1.66e3 - 1.66e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (-1.08e3 + 1.08e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 9.44e3iT - 3.89e7T^{2} \)
83 \( 1 + (911. + 2.19e3i)T + (-3.35e7 + 3.35e7i)T^{2} \)
89 \( 1 + (-952. + 952. i)T - 6.27e7iT^{2} \)
97 \( 1 - 1.85e3T + 8.85e7T^{2} \)
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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

−14.20767662415266096283233400408, −12.96603395286165019470621480428, −11.08719881743150179383769237467, −10.39076725794775098163325182496, −9.805062152657509889657920373182, −8.106784310133429955307011276529, −7.43456776718370621157381267628, −6.40339510118533853843367697012, −3.60848346912361947099412138109, −2.74595720147755349384946952596, 0.06849360849373110634660395894, 1.94227284547967919319673016163, 3.49848728727375079435641124067, 5.69090676910626960621111410457, 7.47024538803838350027401024395, 8.310721927698254812466198783505, 9.182904408044958008722075080521, 9.851994862192395466833828957551, 11.85349237653129631784265028563, 12.58492331731703468071099812531

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