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Properties

Label 2-48-3.2-c26-0-39
Degree $2$
Conductor $48$
Sign $0.990 + 0.136i$
Analytic cond. $205.580$
Root an. cond. $14.3380$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57e6 + 2.17e5i)3-s + 7.56e8i·5-s + 6.43e10·7-s + (2.44e12 + 6.88e11i)9-s − 4.53e13i·11-s − 1.22e14·13-s + (−1.64e14 + 1.19e15i)15-s − 8.44e14i·17-s − 1.38e16·19-s + (1.01e17 + 1.40e16i)21-s + 6.13e17i·23-s + 9.18e17·25-s + (3.71e18 + 1.62e18i)27-s − 1.18e19i·29-s + 2.12e19·31-s + ⋯
L(s)  = 1  + (0.990 + 0.136i)3-s + 0.619i·5-s + 0.664·7-s + (0.962 + 0.270i)9-s − 1.31i·11-s − 0.403·13-s + (−0.0846 + 0.613i)15-s − 0.0852i·17-s − 0.330·19-s + (0.658 + 0.0908i)21-s + 1.21i·23-s + 0.616·25-s + (0.916 + 0.399i)27-s − 1.15i·29-s + 0.869·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.990 + 0.136i$
Analytic conductor: \(205.580\)
Root analytic conductor: \(14.3380\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :13),\ 0.990 + 0.136i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(4.052575344\)
\(L(\frac12)\) \(\approx\) \(4.052575344\)
\(L(14)\) 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 \( 1 + (-1.57e6 - 2.17e5i)T \)
good5 \( 1 - 7.56e8iT - 1.49e18T^{2} \)
7 \( 1 - 6.43e10T + 9.38e21T^{2} \)
11 \( 1 + 4.53e13iT - 1.19e27T^{2} \)
13 \( 1 + 1.22e14T + 9.17e28T^{2} \)
17 \( 1 + 8.44e14iT - 9.81e31T^{2} \)
19 \( 1 + 1.38e16T + 1.76e33T^{2} \)
23 \( 1 - 6.13e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.18e19iT - 1.05e38T^{2} \)
31 \( 1 - 2.12e19T + 5.96e38T^{2} \)
37 \( 1 - 2.40e19T + 5.93e40T^{2} \)
41 \( 1 - 2.62e20iT - 8.55e41T^{2} \)
43 \( 1 - 8.46e20T + 2.95e42T^{2} \)
47 \( 1 + 8.66e21iT - 2.98e43T^{2} \)
53 \( 1 + 3.15e22iT - 6.77e44T^{2} \)
59 \( 1 - 4.46e22iT - 1.10e46T^{2} \)
61 \( 1 + 8.62e22T + 2.62e46T^{2} \)
67 \( 1 - 7.14e23T + 3.00e47T^{2} \)
71 \( 1 + 1.40e24iT - 1.35e48T^{2} \)
73 \( 1 - 1.49e24T + 2.79e48T^{2} \)
79 \( 1 - 1.64e24T + 2.17e49T^{2} \)
83 \( 1 - 1.43e25iT - 7.87e49T^{2} \)
89 \( 1 + 2.88e25iT - 4.83e50T^{2} \)
97 \( 1 - 9.43e25T + 4.52e51T^{2} \)
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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

−10.73423397795203078570781803560, −9.640223107499856962896369662074, −8.478796113237904837206019365028, −7.73164550870495287583223834333, −6.52199069719923603122491253000, −5.11388417013604117043677433468, −3.83130565011354882355507995381, −2.93829416127223614294608140671, −1.97014516819575109685688351140, −0.71682520203553571880265174743, 0.937590876834906945761545943586, 1.85523679823054181479854696733, 2.77752263133767878829728332100, 4.31525071114111843723389886095, 4.87091300579953952337586922859, 6.69284960209477429940024217860, 7.71706873225592670589503211933, 8.620218366595287102923284275951, 9.544205459452503684835849314650, 10.65628157028328799083809750047

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