Properties

Degree $2$
Conductor $48$
Sign $0.730 + 0.682i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.25 − 1.55i)2-s + (1.22 + 1.22i)3-s + (−0.846 − 3.90i)4-s + (0.909 + 0.909i)5-s + (3.44 − 0.368i)6-s − 0.654·7-s + (−7.14 − 3.59i)8-s + 2.99i·9-s + (2.55 − 0.273i)10-s + (−13.3 + 13.3i)11-s + (3.75 − 5.82i)12-s + (8.32 − 8.32i)13-s + (−0.822 + 1.01i)14-s + 2.22i·15-s + (−14.5 + 6.62i)16-s − 3.93·17-s + ⋯
L(s)  = 1  + (0.627 − 0.778i)2-s + (0.408 + 0.408i)3-s + (−0.211 − 0.977i)4-s + (0.181 + 0.181i)5-s + (0.574 − 0.0614i)6-s − 0.0935·7-s + (−0.893 − 0.448i)8-s + 0.333i·9-s + (0.255 − 0.0273i)10-s + (−1.21 + 1.21i)11-s + (0.312 − 0.485i)12-s + (0.640 − 0.640i)13-s + (−0.0587 + 0.0728i)14-s + 0.148i·15-s + (−0.910 + 0.413i)16-s − 0.231·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.730 + 0.682i$
Motivic weight: \(2\)
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.730 + 0.682i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45848 - 0.575358i\)
\(L(\frac12)\) \(\approx\) \(1.45848 - 0.575358i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 + 1.55i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
good5 \( 1 + (-0.909 - 0.909i)T + 25iT^{2} \)
7 \( 1 + 0.654T + 49T^{2} \)
11 \( 1 + (13.3 - 13.3i)T - 121iT^{2} \)
13 \( 1 + (-8.32 + 8.32i)T - 169iT^{2} \)
17 \( 1 + 3.93T + 289T^{2} \)
19 \( 1 + (-16.8 - 16.8i)T + 361iT^{2} \)
23 \( 1 + 23.1T + 529T^{2} \)
29 \( 1 + (-35.6 + 35.6i)T - 841iT^{2} \)
31 \( 1 + 45.5iT - 961T^{2} \)
37 \( 1 + (-10.1 - 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (-22.7 + 22.7i)T - 1.84e3iT^{2} \)
47 \( 1 + 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (-41.5 - 41.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (21.0 - 21.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (68.7 - 68.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-67.8 - 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 33.3T + 5.04e3T^{2} \)
73 \( 1 - 18.6iT - 5.32e3T^{2} \)
79 \( 1 + 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (72.0 + 72.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−15.19657097597148007142780080411, −13.98915678772828762175739871301, −13.07593052938475426779828561993, −11.89591759779988937701865938542, −10.35464272673361048729548389991, −9.838661064819449085706556123930, −7.977134049586329672641170842111, −5.86468027004400497106838915955, −4.31816049767233815026115499319, −2.55531396308722075764088651644, 3.19387372975425808843308269309, 5.23516980961812035077879909926, 6.64371405012618888309592050469, 8.040141682358071671964920248865, 9.046807457792622018594622226546, 11.10196963045264219770057596628, 12.53756573641677571691016584099, 13.59547156347257042232655985693, 14.11333876293954587873895932479, 15.75441409967325005554728754684

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