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Properties

Label 2-48-16.11-c6-0-6
Degree $2$
Conductor $48$
Sign $-0.847 + 0.530i$
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  + (3.36 + 7.25i)2-s + (11.0 + 11.0i)3-s + (−41.3 + 48.8i)4-s + (−26.6 − 26.6i)5-s + (−42.9 + 117. i)6-s − 403.·7-s + (−493. − 135. i)8-s + 242. i·9-s + (103. − 282. i)10-s + (152. − 152. i)11-s + (−994. + 82.7i)12-s + (−1.72e3 + 1.72e3i)13-s + (−1.35e3 − 2.93e3i)14-s − 587. i·15-s + (−676. − 4.03e3i)16-s + 110.·17-s + ⋯
L(s)  = 1  + (0.420 + 0.907i)2-s + (0.408 + 0.408i)3-s + (−0.646 + 0.763i)4-s + (−0.213 − 0.213i)5-s + (−0.198 + 0.542i)6-s − 1.17·7-s + (−0.964 − 0.265i)8-s + 0.333i·9-s + (0.103 − 0.282i)10-s + (0.114 − 0.114i)11-s + (−0.575 + 0.0478i)12-s + (−0.786 + 0.786i)13-s + (−0.495 − 1.06i)14-s − 0.174i·15-s + (−0.165 − 0.986i)16-s + 0.0225·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.847 + 0.530i$
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: \((2,\ 48,\ (\ :3),\ -0.847 + 0.530i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.266067 - 0.927577i\)
\(L(\frac12)\) \(\approx\) \(0.266067 - 0.927577i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.36 - 7.25i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (26.6 + 26.6i)T + 1.56e4iT^{2} \)
7 \( 1 + 403.T + 1.17e5T^{2} \)
11 \( 1 + (-152. + 152. i)T - 1.77e6iT^{2} \)
13 \( 1 + (1.72e3 - 1.72e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 110.T + 2.41e7T^{2} \)
19 \( 1 + (-3.29e3 - 3.29e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 4.14e3T + 1.48e8T^{2} \)
29 \( 1 + (2.62e4 - 2.62e4i)T - 5.94e8iT^{2} \)
31 \( 1 - 1.55e4iT - 8.87e8T^{2} \)
37 \( 1 + (-4.18e4 - 4.18e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 1.35e5iT - 4.75e9T^{2} \)
43 \( 1 + (-2.55e4 + 2.55e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 3.65e4iT - 1.07e10T^{2} \)
53 \( 1 + (-1.15e5 - 1.15e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-7.10e4 + 7.10e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (9.63e4 - 9.63e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (2.66e5 + 2.66e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 4.16e5T + 1.28e11T^{2} \)
73 \( 1 + 2.02e5iT - 1.51e11T^{2} \)
79 \( 1 + 8.32e5iT - 2.43e11T^{2} \)
83 \( 1 + (1.57e5 + 1.57e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 9.85e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.46e6T + 8.32e11T^{2} \)
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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

−15.07049189843474724070873037475, −14.13058144891064939697528189714, −13.01712977267762995347196055027, −11.96218919558581093200098888085, −9.888752124250185300967969676474, −8.954305381378253640535227622426, −7.52644236730338942595189358529, −6.21144047229228809266169930151, −4.56038658525437882899667416029, −3.18957149582239923370416378694, 0.34766617464561315649991132019, 2.49040359489139287902572570268, 3.73398961942240103016219752670, 5.73152301178499868337225017348, 7.36173531290622919542589973004, 9.191389617572781077190314264269, 10.10001850699688572960005100070, 11.56442597784131633368220757493, 12.71199912060983382591701653029, 13.38023631100295410030499489160

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