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Properties

Label 2-48-48.5-c2-0-7
Degree $2$
Conductor $48$
Sign $0.942 - 0.333i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
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  + (1.96 − 0.391i)2-s + (−0.164 + 2.99i)3-s + (3.69 − 1.53i)4-s + (−3.61 + 3.61i)5-s + (0.848 + 5.93i)6-s − 12.2i·7-s + (6.64 − 4.45i)8-s + (−8.94 − 0.985i)9-s + (−5.67 + 8.49i)10-s + (1.76 − 1.76i)11-s + (3.98 + 11.3i)12-s + (−2.38 + 2.38i)13-s + (−4.80 − 24.0i)14-s + (−10.2 − 11.4i)15-s + (11.2 − 11.3i)16-s + 20.0i·17-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)2-s + (−0.0548 + 0.998i)3-s + (0.923 − 0.383i)4-s + (−0.722 + 0.722i)5-s + (0.141 + 0.989i)6-s − 1.75i·7-s + (0.830 − 0.556i)8-s + (−0.993 − 0.109i)9-s + (−0.567 + 0.849i)10-s + (0.160 − 0.160i)11-s + (0.332 + 0.943i)12-s + (−0.183 + 0.183i)13-s + (−0.342 − 1.72i)14-s + (−0.681 − 0.761i)15-s + (0.705 − 0.708i)16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.942 - 0.333i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64485 + 0.282031i\)
\(L(\frac12)\) \(\approx\) \(1.64485 + 0.282031i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 + 0.391i)T \)
3 \( 1 + (0.164 - 2.99i)T \)
good5 \( 1 + (3.61 - 3.61i)T - 25iT^{2} \)
7 \( 1 + 12.2iT - 49T^{2} \)
11 \( 1 + (-1.76 + 1.76i)T - 121iT^{2} \)
13 \( 1 + (2.38 - 2.38i)T - 169iT^{2} \)
17 \( 1 - 20.0iT - 289T^{2} \)
19 \( 1 + (8.77 - 8.77i)T - 361iT^{2} \)
23 \( 1 + 13.1T + 529T^{2} \)
29 \( 1 + (6.51 + 6.51i)T + 841iT^{2} \)
31 \( 1 - 37.5T + 961T^{2} \)
37 \( 1 + (-10.0 - 10.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 4.57T + 1.68e3T^{2} \)
43 \( 1 + (-21.2 - 21.2i)T + 1.84e3iT^{2} \)
47 \( 1 + 54.8iT - 2.20e3T^{2} \)
53 \( 1 + (-21.5 + 21.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (53.6 - 53.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (19.2 - 19.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-31.5 + 31.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 65.1T + 5.04e3T^{2} \)
73 \( 1 + 50.2iT - 5.32e3T^{2} \)
79 \( 1 - 20.9T + 6.24e3T^{2} \)
83 \( 1 + (6.35 + 6.35i)T + 6.88e3iT^{2} \)
89 \( 1 - 166.T + 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−15.20566736024679631241184019593, −14.47267541389709224137529228153, −13.47680354417168425614172026101, −11.80505531400669600146159977183, −10.74711401360872601505472774042, −10.20107762944450518353482012092, −7.78404538589961599335340921909, −6.37407033925700006931156737991, −4.34458299500850320193050611572, −3.57237264357135153619771026819, 2.56525794832904275353494041087, 4.95946913596495597603642220904, 6.21946602275248469617941791187, 7.74907996896149381475953331375, 8.852116239568045571067552910209, 11.51729109169135923024351961111, 12.13740249130370230851163843765, 12.81225400527022107961921123664, 14.14854825545599218044967854132, 15.34745383644204408293850475066

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