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Properties

Label 2-48-16.11-c6-0-7
Degree $2$
Conductor $48$
Sign $0.245 - 0.969i$
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  + (1.27 − 7.89i)2-s + (11.0 + 11.0i)3-s + (−60.7 − 20.1i)4-s + (132. + 132. i)5-s + (101. − 72.9i)6-s − 560.·7-s + (−236. + 453. i)8-s + 242. i·9-s + (1.21e3 − 879. i)10-s + (−1.67e3 + 1.67e3i)11-s + (−447. − 891. i)12-s + (−327. + 327. i)13-s + (−715. + 4.42e3i)14-s + 2.92e3i·15-s + (3.28e3 + 2.44e3i)16-s + 5.11e3·17-s + ⋯
L(s)  = 1  + (0.159 − 0.987i)2-s + (0.408 + 0.408i)3-s + (−0.949 − 0.314i)4-s + (1.06 + 1.06i)5-s + (0.468 − 0.337i)6-s − 1.63·7-s + (−0.462 + 0.886i)8-s + 0.333i·9-s + (1.21 − 0.879i)10-s + (−1.25 + 1.25i)11-s + (−0.258 − 0.516i)12-s + (−0.148 + 0.148i)13-s + (−0.260 + 1.61i)14-s + 0.867i·15-s + (0.801 + 0.597i)16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.245 - 0.969i$
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: \((2,\ 48,\ (\ :3),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.00638 + 0.783217i\)
\(L(\frac12)\) \(\approx\) \(1.00638 + 0.783217i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 + 7.89i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (-132. - 132. i)T + 1.56e4iT^{2} \)
7 \( 1 + 560.T + 1.17e5T^{2} \)
11 \( 1 + (1.67e3 - 1.67e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (327. - 327. i)T - 4.82e6iT^{2} \)
17 \( 1 - 5.11e3T + 2.41e7T^{2} \)
19 \( 1 + (3.62e3 + 3.62e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.59e4T + 1.48e8T^{2} \)
29 \( 1 + (1.71e4 - 1.71e4i)T - 5.94e8iT^{2} \)
31 \( 1 - 2.14e4iT - 8.87e8T^{2} \)
37 \( 1 + (1.51e4 + 1.51e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 4.89e4iT - 4.75e9T^{2} \)
43 \( 1 + (3.73e4 - 3.73e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 9.96e4iT - 1.07e10T^{2} \)
53 \( 1 + (-2.51e4 - 2.51e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (2.35e5 - 2.35e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.37e5 + 1.37e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-1.54e5 - 1.54e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 4.56e5T + 1.28e11T^{2} \)
73 \( 1 - 4.79e5iT - 1.51e11T^{2} \)
79 \( 1 + 4.92e4iT - 2.43e11T^{2} \)
83 \( 1 + (3.13e5 + 3.13e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 6.97e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.88e5T + 8.32e11T^{2} \)
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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

−14.45989036980022580953676195504, −13.28052117353382394105377730584, −12.63463255000912777778284516400, −10.61045694605677626465078668130, −10.06393388712687695289689311124, −9.254778829396140212557094228388, −6.98240130497991679749864663931, −5.32271748479616992705147015136, −3.26746595828227611334421955444, −2.37131884721256526017755562116, 0.50053016686565330694973214860, 3.17488330842947539479554855206, 5.42373989094173540743243585689, 6.26666476919219924929937237490, 7.950475978989428063666693797646, 9.108011587680347079008774718682, 9.986355318007250071440711481136, 12.76099015306858108616574117162, 13.05894153177548531618911666479, 13.90147896145148311985271485483

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