Properties

Degree 2
Conductor $ 2^{4} \cdot 3 $
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

\( d \)  =  \(2\)
\( N \)  =  \(48\)    =    \(2^{4} \cdot 3\)
\( \varepsilon \)  =  $0.730 - 0.682i$
motivic weight  =  \(2\)
character  :  $\chi_{48} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 48,\ (\ :1),\ 0.730 - 0.682i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.45848 + 0.575358i\)
\(L(\frac12)\)  \(\approx\)  \(1.45848 + 0.575358i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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