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Properties

Label 2-48-16.3-c6-0-10
Degree $2$
Conductor $48$
Sign $0.964 + 0.263i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.29 + 3.29i)2-s + (11.0 − 11.0i)3-s + (42.3 − 48.0i)4-s + (−73.3 + 73.3i)5-s + (−44.0 + 116. i)6-s + 291.·7-s + (−150. + 489. i)8-s − 242. i·9-s + (293. − 776. i)10-s + (−1.59e3 − 1.59e3i)11-s + (−62.7 − 995. i)12-s + (1.76e3 + 1.76e3i)13-s + (−2.12e3 + 959. i)14-s + 1.61e3i·15-s + (−513. − 4.06e3i)16-s + 6.75e3·17-s + ⋯
L(s)  = 1  + (−0.911 + 0.411i)2-s + (0.408 − 0.408i)3-s + (0.661 − 0.750i)4-s + (−0.587 + 0.587i)5-s + (−0.204 + 0.540i)6-s + 0.849·7-s + (−0.293 + 0.955i)8-s − 0.333i·9-s + (0.293 − 0.776i)10-s + (−1.19 − 1.19i)11-s + (−0.0362 − 0.576i)12-s + (0.804 + 0.804i)13-s + (−0.774 + 0.349i)14-s + 0.479i·15-s + (−0.125 − 0.992i)16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ 0.964 + 0.263i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.28255 - 0.172182i\)
\(L(\frac12)\) \(\approx\) \(1.28255 - 0.172182i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.29 - 3.29i)T \)
3 \( 1 + (-11.0 + 11.0i)T \)
good5 \( 1 + (73.3 - 73.3i)T - 1.56e4iT^{2} \)
7 \( 1 - 291.T + 1.17e5T^{2} \)
11 \( 1 + (1.59e3 + 1.59e3i)T + 1.77e6iT^{2} \)
13 \( 1 + (-1.76e3 - 1.76e3i)T + 4.82e6iT^{2} \)
17 \( 1 - 6.75e3T + 2.41e7T^{2} \)
19 \( 1 + (-8.59e3 + 8.59e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 1.61e4T + 1.48e8T^{2} \)
29 \( 1 + (7.77e3 + 7.77e3i)T + 5.94e8iT^{2} \)
31 \( 1 + 1.84e4iT - 8.87e8T^{2} \)
37 \( 1 + (-5.21e4 + 5.21e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 3.93e4iT - 4.75e9T^{2} \)
43 \( 1 + (-4.81e3 - 4.81e3i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.11e4iT - 1.07e10T^{2} \)
53 \( 1 + (1.03e5 - 1.03e5i)T - 2.21e10iT^{2} \)
59 \( 1 + (-1.52e5 - 1.52e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (7.55e4 + 7.55e4i)T + 5.15e10iT^{2} \)
67 \( 1 + (3.37e5 - 3.37e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 3.06e5T + 1.28e11T^{2} \)
73 \( 1 - 1.66e5iT - 1.51e11T^{2} \)
79 \( 1 - 1.31e5iT - 2.43e11T^{2} \)
83 \( 1 + (-3.84e5 + 3.84e5i)T - 3.26e11iT^{2} \)
89 \( 1 - 3.79e5iT - 4.96e11T^{2} \)
97 \( 1 + 5.77e5T + 8.32e11T^{2} \)
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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

−14.54529446911343912977810826035, −13.52194953656049442789814403102, −11.45808790137156051517892864917, −10.96025538025918215269804929245, −9.198516795787533902083403644301, −8.005773950080906876054503904962, −7.26980341455722674298730527293, −5.55892464932051993669312643368, −2.94595729092494195830140814874, −0.939332553778342005740005279179, 1.26260084768307556880284915021, 3.24533683499023504861797515681, 5.04308334890384755428194008509, 7.75220617498804777255097774503, 8.128088117564910596503093798632, 9.697702327197756700021768725485, 10.66457373549496802585786284303, 11.97528436202534203853086194090, 12.96785770613423466975409929031, 14.77175225759221387911823394969

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