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Properties

Label 2-48-16.11-c6-0-11
Degree $2$
Conductor $48$
Sign $0.908 + 0.418i$
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  + (−7.86 + 1.48i)2-s + (−11.0 − 11.0i)3-s + (59.5 − 23.3i)4-s + (52.2 + 52.2i)5-s + (103. + 70.2i)6-s − 282.·7-s + (−433. + 271. i)8-s + 242. i·9-s + (−487. − 332. i)10-s + (−135. + 135. i)11-s + (−914. − 399. i)12-s + (730. − 730. i)13-s + (2.22e3 − 419. i)14-s − 1.15e3i·15-s + (3.00e3 − 2.78e3i)16-s + 8.44e3·17-s + ⋯
L(s)  = 1  + (−0.982 + 0.185i)2-s + (−0.408 − 0.408i)3-s + (0.931 − 0.364i)4-s + (0.417 + 0.417i)5-s + (0.476 + 0.325i)6-s − 0.823·7-s + (−0.847 + 0.530i)8-s + 0.333i·9-s + (−0.487 − 0.332i)10-s + (−0.102 + 0.102i)11-s + (−0.528 − 0.231i)12-s + (0.332 − 0.332i)13-s + (0.809 − 0.152i)14-s − 0.341i·15-s + (0.734 − 0.678i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.908 + 0.418i$
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: \((2,\ 48,\ (\ :3),\ 0.908 + 0.418i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.925985 - 0.203078i\)
\(L(\frac12)\) \(\approx\) \(0.925985 - 0.203078i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.86 - 1.48i)T \)
3 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (-52.2 - 52.2i)T + 1.56e4iT^{2} \)
7 \( 1 + 282.T + 1.17e5T^{2} \)
11 \( 1 + (135. - 135. i)T - 1.77e6iT^{2} \)
13 \( 1 + (-730. + 730. i)T - 4.82e6iT^{2} \)
17 \( 1 - 8.44e3T + 2.41e7T^{2} \)
19 \( 1 + (1.83e3 + 1.83e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.44e4T + 1.48e8T^{2} \)
29 \( 1 + (-2.25e4 + 2.25e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 2.57e4iT - 8.87e8T^{2} \)
37 \( 1 + (-2.26e4 - 2.26e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 1.85e4iT - 4.75e9T^{2} \)
43 \( 1 + (-1.82e4 + 1.82e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 4.38e4iT - 1.07e10T^{2} \)
53 \( 1 + (-1.50e5 - 1.50e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-1.69e5 + 1.69e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-2.12e5 + 2.12e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-6.01e4 - 6.01e4i)T + 9.04e10iT^{2} \)
71 \( 1 + 4.74e5T + 1.28e11T^{2} \)
73 \( 1 - 6.41e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.15e5iT - 2.43e11T^{2} \)
83 \( 1 + (1.79e5 + 1.79e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 5.93e5iT - 4.96e11T^{2} \)
97 \( 1 - 7.72e5T + 8.32e11T^{2} \)
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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

−14.50158471805672234272469180876, −13.03276349297056113377132446943, −11.79903709673497001430831534121, −10.48579712852360146624447197694, −9.653159154665385934995577517872, −8.055921560940539312402116474949, −6.76563181758849372107389462834, −5.77298134169797176590423177626, −2.77171436617759132975178506186, −0.811339703024130958400606883367, 1.06731886337082017230809704413, 3.28152035275494220408566219905, 5.59758811185479799122687699661, 6.98202910277467852651642875011, 8.688000344429073072551521599317, 9.711842164315985428451254698424, 10.61678375945401941516181098830, 11.98514544443780320768731461709, 12.99009806834878508874376054342, 14.75534366515186208855725659027

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