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Properties

Label 2-48-48.29-c2-0-3
Degree $2$
Conductor $48$
Sign $0.647 - 0.762i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 + 1.08i)2-s + (2.90 + 0.737i)3-s + (1.64 − 3.64i)4-s + (1.57 + 1.57i)5-s + (−5.68 + 1.91i)6-s + 3.64i·7-s + (1.19 + 7.91i)8-s + (7.91 + 4.29i)9-s + (−4.35 − 0.937i)10-s + (1.19 + 1.19i)11-s + (7.47 − 9.38i)12-s + (−14.6 − 14.6i)13-s + (−3.95 − 6.12i)14-s + (3.41 + 5.74i)15-s + (−10.5 − 12.0i)16-s − 28.0i·17-s + ⋯
L(s)  = 1  + (−0.840 + 0.542i)2-s + (0.969 + 0.245i)3-s + (0.411 − 0.911i)4-s + (0.314 + 0.314i)5-s + (−0.947 + 0.319i)6-s + 0.520i·7-s + (0.148 + 0.988i)8-s + (0.878 + 0.476i)9-s + (−0.435 − 0.0937i)10-s + (0.108 + 0.108i)11-s + (0.622 − 0.782i)12-s + (−1.12 − 1.12i)13-s + (−0.282 − 0.437i)14-s + (0.227 + 0.382i)15-s + (−0.661 − 0.750i)16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.647 - 0.762i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ 0.647 - 0.762i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.949991 + 0.439664i\)
\(L(\frac12)\) \(\approx\) \(0.949991 + 0.439664i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.68 - 1.08i)T \)
3 \( 1 + (-2.90 - 0.737i)T \)
good5 \( 1 + (-1.57 - 1.57i)T + 25iT^{2} \)
7 \( 1 - 3.64iT - 49T^{2} \)
11 \( 1 + (-1.19 - 1.19i)T + 121iT^{2} \)
13 \( 1 + (14.6 + 14.6i)T + 169iT^{2} \)
17 \( 1 + 28.0iT - 289T^{2} \)
19 \( 1 + (-12.5 - 12.5i)T + 361iT^{2} \)
23 \( 1 + 29.2T + 529T^{2} \)
29 \( 1 + (19.3 - 19.3i)T - 841iT^{2} \)
31 \( 1 - 11.6T + 961T^{2} \)
37 \( 1 + (-0.771 + 0.771i)T - 1.36e3iT^{2} \)
41 \( 1 + 25.6T + 1.68e3T^{2} \)
43 \( 1 + (40.5 - 40.5i)T - 1.84e3iT^{2} \)
47 \( 1 + 50.2iT - 2.20e3T^{2} \)
53 \( 1 + (-46.2 - 46.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (-22.7 - 22.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-12.7 - 12.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (-10.6 - 10.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 122.T + 5.04e3T^{2} \)
73 \( 1 - 15.0iT - 5.32e3T^{2} \)
79 \( 1 + 51.3T + 6.24e3T^{2} \)
83 \( 1 + (37.8 - 37.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 5.45T + 7.92e3T^{2} \)
97 \( 1 + 81.1T + 9.40e3T^{2} \)
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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

−15.55773285595372195427022412148, −14.65939761710809094876527704010, −13.80448291556302285320825902979, −12.01488124506223389876914981053, −10.19649698214319714409273471053, −9.588963290891858445433794600384, −8.227243440540683612655237901280, −7.17379882616971866445988525844, −5.31427078937959112917603791232, −2.55914758882068127390786325796, 1.92480159761492582684069068044, 3.92938028915399901188281093598, 6.90300451541053274614993403227, 8.086425040202282000609500031867, 9.286383147738615666092538113438, 10.13762696656707314638999060601, 11.77528738461065206848360601442, 12.93522915643596283802512320842, 13.95525412959094064185398121178, 15.30489415069490792631914401955

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