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Properties

Label 2-48-16.11-c6-0-10
Degree $2$
Conductor $48$
Sign $0.205 - 0.978i$
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  + (−7.91 − 1.19i)2-s + (11.0 + 11.0i)3-s + (61.1 + 18.9i)4-s + (156. + 156. i)5-s + (−74.0 − 100. i)6-s + 23.3·7-s + (−461. − 222. i)8-s + 242. i·9-s + (−1.04e3 − 1.42e3i)10-s + (1.21e3 − 1.21e3i)11-s + (465. + 882. i)12-s + (−1.85e3 + 1.85e3i)13-s + (−184. − 27.9i)14-s + 3.44e3i·15-s + (3.38e3 + 2.31e3i)16-s − 445.·17-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (0.408 + 0.408i)3-s + (0.955 + 0.295i)4-s + (1.25 + 1.25i)5-s + (−0.342 − 0.464i)6-s + 0.0681·7-s + (−0.900 − 0.434i)8-s + 0.333i·9-s + (−1.04 − 1.42i)10-s + (0.910 − 0.910i)11-s + (0.269 + 0.510i)12-s + (−0.842 + 0.842i)13-s + (−0.0673 − 0.0101i)14-s + 1.02i·15-s + (0.825 + 0.564i)16-s − 0.0907·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ 0.205 - 0.978i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.21874 + 0.989371i\)
\(L(\frac12)\) \(\approx\) \(1.21874 + 0.989371i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.91 + 1.19i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (-156. - 156. i)T + 1.56e4iT^{2} \)
7 \( 1 - 23.3T + 1.17e5T^{2} \)
11 \( 1 + (-1.21e3 + 1.21e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (1.85e3 - 1.85e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 445.T + 2.41e7T^{2} \)
19 \( 1 + (-5.08e3 - 5.08e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 5.62e3T + 1.48e8T^{2} \)
29 \( 1 + (-1.65e4 + 1.65e4i)T - 5.94e8iT^{2} \)
31 \( 1 - 4.88e4iT - 8.87e8T^{2} \)
37 \( 1 + (5.29e4 + 5.29e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 9.48e3iT - 4.75e9T^{2} \)
43 \( 1 + (-3.22e4 + 3.22e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 6.36e4iT - 1.07e10T^{2} \)
53 \( 1 + (-1.26e5 - 1.26e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-1.15e5 + 1.15e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.82e5 - 1.82e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-9.34e4 - 9.34e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 5.32e5T + 1.28e11T^{2} \)
73 \( 1 + 4.93e5iT - 1.51e11T^{2} \)
79 \( 1 + 7.44e5iT - 2.43e11T^{2} \)
83 \( 1 + (3.87e5 + 3.87e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.17e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.27e6T + 8.32e11T^{2} \)
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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

−14.44117919164124914653643412133, −14.02312365095051679923143123286, −11.92436457358663934568290904349, −10.69223450735508669153832771631, −9.838640479392979072221218060528, −8.907211277752052666372791278226, −7.18251068001413611684768212005, −6.05334138961048724000845290000, −3.23577794900611810699076563739, −1.86630071552388454452978101308, 0.976882018800167297668005891487, 2.21761685093339768279855155262, 5.21824452120097880823172408694, 6.72985571807146408614161386870, 8.178449571218769796856821395292, 9.364103100163974440815831704147, 9.928090777004883877458442797948, 11.92568135745968132323415516181, 12.89761915040101567394332532250, 14.19863633725529943447508048554

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