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Properties

Label 2-48-16.5-c3-0-10
Degree $2$
Conductor $48$
Sign $-0.832 + 0.554i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
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  + (0.954 − 2.66i)2-s + (−2.12 + 2.12i)3-s + (−6.17 − 5.08i)4-s + (−8.83 − 8.83i)5-s + (3.62 + 7.67i)6-s − 29.4i·7-s + (−19.4 + 11.6i)8-s − 8.99i·9-s + (−31.9 + 15.0i)10-s + (44.6 + 44.6i)11-s + (23.8 − 2.33i)12-s + (6.83 − 6.83i)13-s + (−78.4 − 28.1i)14-s + 37.4·15-s + (12.3 + 62.7i)16-s − 56.8·17-s + ⋯
L(s)  = 1  + (0.337 − 0.941i)2-s + (−0.408 + 0.408i)3-s + (−0.772 − 0.635i)4-s + (−0.790 − 0.790i)5-s + (0.246 + 0.522i)6-s − 1.59i·7-s + (−0.858 + 0.513i)8-s − 0.333i·9-s + (−1.01 + 0.477i)10-s + (1.22 + 1.22i)11-s + (0.574 − 0.0560i)12-s + (0.145 − 0.145i)13-s + (−1.49 − 0.536i)14-s + 0.645·15-s + (0.193 + 0.981i)16-s − 0.810·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.297232 - 0.982872i\)
\(L(\frac12)\) \(\approx\) \(0.297232 - 0.982872i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.954 + 2.66i)T \)
3 \( 1 + (2.12 - 2.12i)T \)
good5 \( 1 + (8.83 + 8.83i)T + 125iT^{2} \)
7 \( 1 + 29.4iT - 343T^{2} \)
11 \( 1 + (-44.6 - 44.6i)T + 1.33e3iT^{2} \)
13 \( 1 + (-6.83 + 6.83i)T - 2.19e3iT^{2} \)
17 \( 1 + 56.8T + 4.91e3T^{2} \)
19 \( 1 + (-91.0 + 91.0i)T - 6.85e3iT^{2} \)
23 \( 1 + 96.8iT - 1.21e4T^{2} \)
29 \( 1 + (59.2 - 59.2i)T - 2.43e4iT^{2} \)
31 \( 1 - 103.T + 2.97e4T^{2} \)
37 \( 1 + (79.5 + 79.5i)T + 5.06e4iT^{2} \)
41 \( 1 + 105. iT - 6.89e4T^{2} \)
43 \( 1 + (-39.9 - 39.9i)T + 7.95e4iT^{2} \)
47 \( 1 - 9.34T + 1.03e5T^{2} \)
53 \( 1 + (-245. - 245. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-345. - 345. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-370. + 370. i)T - 2.26e5iT^{2} \)
67 \( 1 + (595. - 595. i)T - 3.00e5iT^{2} \)
71 \( 1 - 493. iT - 3.57e5T^{2} \)
73 \( 1 - 33.2iT - 3.89e5T^{2} \)
79 \( 1 - 552.T + 4.93e5T^{2} \)
83 \( 1 + (18.5 - 18.5i)T - 5.71e5iT^{2} \)
89 \( 1 + 934. iT - 7.04e5T^{2} \)
97 \( 1 - 1.42e3T + 9.12e5T^{2} \)
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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

−14.52767608104035655353553625819, −13.33122842033236124731521564687, −12.20095215139785282908517665462, −11.28070650603621926892396125151, −10.13957035922908904185670565301, −8.960873734764122897493073865412, −7.02212938633260898707140094839, −4.66888748583498289071487256812, −4.01794145490105090972375785924, −0.813976544335131458642851322134, 3.47879704841021074834349220712, 5.63477910813846850059573883421, 6.59304002942363500662255214135, 8.034682476076494404349076091556, 9.172821153862196590197204534745, 11.54201725909688337735505781031, 11.96711282054043006993395602546, 13.57276243719974423364167983730, 14.68021297842820223764344357844, 15.58221992571751883930968970744

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