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Properties

Label 2-48-16.5-c3-0-1
Degree $2$
Conductor $48$
Sign $0.618 - 0.785i$
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  + (−1.40 − 2.45i)2-s + (−2.12 + 2.12i)3-s + (−4.04 + 6.90i)4-s + (3.22 + 3.22i)5-s + (8.18 + 2.22i)6-s + 24.6i·7-s + (22.6 + 0.211i)8-s − 8.99i·9-s + (3.37 − 12.4i)10-s + (23.7 + 23.7i)11-s + (−6.06 − 23.2i)12-s + (−18.6 + 18.6i)13-s + (60.3 − 34.6i)14-s − 13.6·15-s + (−31.3 − 55.8i)16-s − 3.55·17-s + ⋯
L(s)  = 1  + (−0.497 − 0.867i)2-s + (−0.408 + 0.408i)3-s + (−0.505 + 0.862i)4-s + (0.288 + 0.288i)5-s + (0.557 + 0.151i)6-s + 1.32i·7-s + (0.999 + 0.00936i)8-s − 0.333i·9-s + (0.106 − 0.393i)10-s + (0.651 + 0.651i)11-s + (−0.145 − 0.558i)12-s + (−0.398 + 0.398i)13-s + (1.15 − 0.660i)14-s − 0.235·15-s + (−0.489 − 0.872i)16-s − 0.0506·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.618 - 0.785i$
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: \((2,\ 48,\ (\ :3/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.745750 + 0.361995i\)
\(L(\frac12)\) \(\approx\) \(0.745750 + 0.361995i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 2.45i)T \)
3 \( 1 + (2.12 - 2.12i)T \)
good5 \( 1 + (-3.22 - 3.22i)T + 125iT^{2} \)
7 \( 1 - 24.6iT - 343T^{2} \)
11 \( 1 + (-23.7 - 23.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (18.6 - 18.6i)T - 2.19e3iT^{2} \)
17 \( 1 + 3.55T + 4.91e3T^{2} \)
19 \( 1 + (109. - 109. i)T - 6.85e3iT^{2} \)
23 \( 1 + 36.5iT - 1.21e4T^{2} \)
29 \( 1 + (-68.8 + 68.8i)T - 2.43e4iT^{2} \)
31 \( 1 - 306.T + 2.97e4T^{2} \)
37 \( 1 + (-92.9 - 92.9i)T + 5.06e4iT^{2} \)
41 \( 1 + 385. iT - 6.89e4T^{2} \)
43 \( 1 + (-150. - 150. i)T + 7.95e4iT^{2} \)
47 \( 1 + 114.T + 1.03e5T^{2} \)
53 \( 1 + (-451. - 451. i)T + 1.48e5iT^{2} \)
59 \( 1 + (544. + 544. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-179. + 179. i)T - 2.26e5iT^{2} \)
67 \( 1 + (283. - 283. i)T - 3.00e5iT^{2} \)
71 \( 1 + 930. iT - 3.57e5T^{2} \)
73 \( 1 - 701. iT - 3.89e5T^{2} \)
79 \( 1 - 779.T + 4.93e5T^{2} \)
83 \( 1 + (296. - 296. i)T - 5.71e5iT^{2} \)
89 \( 1 - 865. iT - 7.04e5T^{2} \)
97 \( 1 + 542.T + 9.12e5T^{2} \)
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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

−15.37224535348432962481729498287, −14.19119367972926173040669825999, −12.38515051054971363071382314052, −11.95681705487884624463411589505, −10.48618245091943970013782973490, −9.539091907954741262706117179311, −8.381990700029690853231044459093, −6.30901449463204588505969749120, −4.40603699311973756928159018010, −2.30293058000477669032709386687, 0.835139178902662331150660379400, 4.62810602663029688101435360330, 6.28240020188754860599966046635, 7.31655574329489187955159764253, 8.693326323759231696278544217702, 10.12703612693932687486168986360, 11.20515924775881695976794195532, 13.11958892267793960152598077743, 13.82637869241002386200321070169, 15.10603249312558997041237830273

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