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Properties

Label 2-48-16.3-c8-0-23
Degree $2$
Conductor $48$
Sign $0.180 + 0.983i$
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  + (−9.57 + 12.8i)2-s + (33.0 − 33.0i)3-s + (−72.5 − 245. i)4-s + (541. − 541. i)5-s + (107. + 740. i)6-s + 1.14e3·7-s + (3.84e3 + 1.42e3i)8-s − 2.18e3i·9-s + (1.75e3 + 1.21e4i)10-s + (−8.23e3 − 8.23e3i)11-s + (−1.05e4 − 5.71e3i)12-s + (−8.05e3 − 8.05e3i)13-s + (−1.09e4 + 1.46e4i)14-s − 3.58e4i·15-s + (−5.50e4 + 3.56e4i)16-s − 8.47e4·17-s + ⋯
L(s)  = 1  + (−0.598 + 0.801i)2-s + (0.408 − 0.408i)3-s + (−0.283 − 0.959i)4-s + (0.867 − 0.867i)5-s + (0.0826 + 0.571i)6-s + 0.477·7-s + (0.937 + 0.347i)8-s − 0.333i·9-s + (0.175 + 1.21i)10-s + (−0.562 − 0.562i)11-s + (−0.507 − 0.275i)12-s + (−0.282 − 0.282i)13-s + (−0.285 + 0.382i)14-s − 0.707i·15-s + (−0.839 + 0.543i)16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(19.5541\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :4),\ 0.180 + 0.983i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.14550 - 0.954057i\)
\(L(\frac12)\) \(\approx\) \(1.14550 - 0.954057i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.57 - 12.8i)T \)
3 \( 1 + (-33.0 + 33.0i)T \)
good5 \( 1 + (-541. + 541. i)T - 3.90e5iT^{2} \)
7 \( 1 - 1.14e3T + 5.76e6T^{2} \)
11 \( 1 + (8.23e3 + 8.23e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (8.05e3 + 8.05e3i)T + 8.15e8iT^{2} \)
17 \( 1 + 8.47e4T + 6.97e9T^{2} \)
19 \( 1 + (-3.33e4 + 3.33e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 9.17e4T + 7.83e10T^{2} \)
29 \( 1 + (-3.36e4 - 3.36e4i)T + 5.00e11iT^{2} \)
31 \( 1 + 1.21e6iT - 8.52e11T^{2} \)
37 \( 1 + (-1.95e6 + 1.95e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 6.97e5iT - 7.98e12T^{2} \)
43 \( 1 + (3.81e6 + 3.81e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 8.51e6iT - 2.38e13T^{2} \)
53 \( 1 + (-1.54e6 + 1.54e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (4.42e6 + 4.42e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (3.48e5 + 3.48e5i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.95e7 + 1.95e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 1.94e7T + 6.45e14T^{2} \)
73 \( 1 - 2.88e7iT - 8.06e14T^{2} \)
79 \( 1 + 7.21e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.45e7 + 3.45e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 7.91e7iT - 3.93e15T^{2} \)
97 \( 1 - 5.29e7T + 7.83e15T^{2} \)
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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

−13.68908042876369064499061901111, −13.00066053994921396411095618079, −11.06866912509284787927442484654, −9.592939114562165172569314905121, −8.700101179295770421881969163153, −7.62568750692571674546842339784, −6.06063112682542341092441209681, −4.88949770012636073238229502869, −2.05185378604278545517873956244, −0.61732372180713024062780724619, 1.84662958900520808451472322443, 2.92964449652115443596910998364, 4.72166303418826530939380311757, 6.91598881321267539609797846476, 8.352533072879222149422695832827, 9.675573695267220131500700097181, 10.42029955043417249696286774487, 11.49353084144499991721201313014, 13.05567411885464409356812608296, 14.05453767629669942325201523890

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