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Properties

Label 2-48-16.11-c8-0-2
Degree $2$
Conductor $48$
Sign $-0.202 + 0.979i$
Analytic cond. $19.5541$
Root an. cond. $4.42201$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.37 + 15.8i)2-s + (33.0 + 33.0i)3-s + (−244. − 75.2i)4-s + (508. + 508. i)5-s + (−601. + 444. i)6-s − 3.25e3·7-s + (1.77e3 − 3.69e3i)8-s + 2.18e3i·9-s + (−9.25e3 + 6.83e3i)10-s + (6.77e3 − 6.77e3i)11-s + (−5.60e3 − 1.05e4i)12-s + (−2.60e4 + 2.60e4i)13-s + (7.73e3 − 5.14e4i)14-s + 3.36e4i·15-s + (5.42e4 + 3.68e4i)16-s − 1.55e5·17-s + ⋯
L(s)  = 1  + (−0.148 + 0.988i)2-s + (0.408 + 0.408i)3-s + (−0.955 − 0.293i)4-s + (0.813 + 0.813i)5-s + (−0.464 + 0.343i)6-s − 1.35·7-s + (0.432 − 0.901i)8-s + 0.333i·9-s + (−0.925 + 0.683i)10-s + (0.462 − 0.462i)11-s + (−0.270 − 0.510i)12-s + (−0.912 + 0.912i)13-s + (0.201 − 1.33i)14-s + 0.664i·15-s + (0.827 + 0.561i)16-s − 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.202 + 0.979i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ -0.202 + 0.979i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.258766 - 0.317779i\)
\(L(\frac12)\) \(\approx\) \(0.258766 - 0.317779i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.37 - 15.8i)T \)
3 \( 1 + (-33.0 - 33.0i)T \)
good5 \( 1 + (-508. - 508. i)T + 3.90e5iT^{2} \)
7 \( 1 + 3.25e3T + 5.76e6T^{2} \)
11 \( 1 + (-6.77e3 + 6.77e3i)T - 2.14e8iT^{2} \)
13 \( 1 + (2.60e4 - 2.60e4i)T - 8.15e8iT^{2} \)
17 \( 1 + 1.55e5T + 6.97e9T^{2} \)
19 \( 1 + (9.81e4 + 9.81e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 4.40e5T + 7.83e10T^{2} \)
29 \( 1 + (-4.80e5 + 4.80e5i)T - 5.00e11iT^{2} \)
31 \( 1 + 6.27e5iT - 8.52e11T^{2} \)
37 \( 1 + (1.48e6 + 1.48e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 4.23e4iT - 7.98e12T^{2} \)
43 \( 1 + (1.53e6 - 1.53e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 8.84e6iT - 2.38e13T^{2} \)
53 \( 1 + (6.41e6 + 6.41e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (9.15e6 - 9.15e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (1.53e7 - 1.53e7i)T - 1.91e14iT^{2} \)
67 \( 1 + (-9.64e6 - 9.64e6i)T + 4.06e14iT^{2} \)
71 \( 1 + 9.00e6T + 6.45e14T^{2} \)
73 \( 1 - 1.17e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.38e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.31e7 - 1.31e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 6.91e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.04e8T + 7.83e15T^{2} \)
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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

−14.84226206984601277796741564159, −13.81235820841265275287393203213, −13.03568859733618650710859304955, −10.81321999090313791822158856646, −9.543186627314940620867730304134, −8.993411882747482614802682630642, −6.89161497643103310626942621967, −6.34589030793569009830890393742, −4.43342108171722526545578258803, −2.66761599626081650512791153142, 0.14133434280325421628867876183, 1.74670969957525242211836149333, 3.10203864654472439609079033338, 4.90236910263398646106098082870, 6.72880471427466799361125466177, 8.691175543964757682532593305534, 9.433146499153180407956215367862, 10.48113332347117179075327795175, 12.39909601772521500756139444225, 12.86522748125627001071216753569

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