Properties

Degree $2$
Conductor $48$
Sign $0.901 - 0.433i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.96 + 0.362i)2-s + (−1.22 + 1.22i)3-s + (3.73 + 1.42i)4-s + (1.69 − 1.69i)5-s + (−2.85 + 1.96i)6-s − 5.74·7-s + (6.83 + 4.16i)8-s − 2.99i·9-s + (3.95 − 2.72i)10-s + (−5.59 − 5.59i)11-s + (−6.32 + 2.82i)12-s + (−13.5 − 13.5i)13-s + (−11.2 − 2.08i)14-s + 4.16i·15-s + (11.9 + 10.6i)16-s + 19.7·17-s + ⋯
L(s)  = 1  + (0.983 + 0.181i)2-s + (−0.408 + 0.408i)3-s + (0.934 + 0.356i)4-s + (0.339 − 0.339i)5-s + (−0.475 + 0.327i)6-s − 0.820·7-s + (0.853 + 0.520i)8-s − 0.333i·9-s + (0.395 − 0.272i)10-s + (−0.508 − 0.508i)11-s + (−0.527 + 0.235i)12-s + (−1.04 − 1.04i)13-s + (−0.806 − 0.148i)14-s + 0.277i·15-s + (0.745 + 0.666i)16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.901 - 0.433i$
Motivic weight: \(2\)
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.901 - 0.433i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59659 + 0.363853i\)
\(L(\frac12)\) \(\approx\) \(1.59659 + 0.363853i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 - 0.362i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
good5 \( 1 + (-1.69 + 1.69i)T - 25iT^{2} \)
7 \( 1 + 5.74T + 49T^{2} \)
11 \( 1 + (5.59 + 5.59i)T + 121iT^{2} \)
13 \( 1 + (13.5 + 13.5i)T + 169iT^{2} \)
17 \( 1 - 19.7T + 289T^{2} \)
19 \( 1 + (21.6 - 21.6i)T - 361iT^{2} \)
23 \( 1 - 24.9T + 529T^{2} \)
29 \( 1 + (-1.50 - 1.50i)T + 841iT^{2} \)
31 \( 1 + 2.20iT - 961T^{2} \)
37 \( 1 + (-27.6 + 27.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 51.3iT - 1.68e3T^{2} \)
43 \( 1 + (-21.4 - 21.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 76.5iT - 2.20e3T^{2} \)
53 \( 1 + (56.5 - 56.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (48.0 + 48.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (51.5 + 51.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (-63.4 + 63.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 43.4T + 5.04e3T^{2} \)
73 \( 1 - 73.9iT - 5.32e3T^{2} \)
79 \( 1 + 4.12iT - 6.24e3T^{2} \)
83 \( 1 + (-38.4 + 38.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 52.9iT - 7.92e3T^{2} \)
97 \( 1 - 23.1T + 9.40e3T^{2} \)
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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

−15.41031300408144267539063991886, −14.48488442463013873169892010992, −12.95785342126518948722900194360, −12.48654151409225974974410063771, −10.88780673221453439793642281130, −9.803309952228221753038022453490, −7.81693037772378272542497007187, −6.12270639760079213565485984484, −5.10956121571638227167002534367, −3.22780090768164610083624752412, 2.57643030537763233134342356400, 4.76261107057091634498132853322, 6.32861190839742863638779117048, 7.25499018460813072536822610696, 9.701353319833897262032172292097, 10.84849631354856949048946790229, 12.19903433499093960689902427778, 12.92747767943381285503393531944, 14.08309648786229992221436811023, 15.12671167483175888391269655965

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