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Properties

Label 2-48-16.5-c3-0-9
Degree $2$
Conductor $48$
Sign $0.245 + 0.969i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.07 − 1.92i)2-s + (2.12 − 2.12i)3-s + (0.577 − 7.97i)4-s + (0.644 + 0.644i)5-s + (0.306 − 8.47i)6-s + 7.13i·7-s + (−14.1 − 17.6i)8-s − 8.99i·9-s + (2.57 + 0.0932i)10-s + (25.4 + 25.4i)11-s + (−15.7 − 18.1i)12-s + (−14.6 + 14.6i)13-s + (13.7 + 14.7i)14-s + 2.73·15-s + (−63.3 − 9.22i)16-s + 71.4·17-s + ⋯
L(s)  = 1  + (0.732 − 0.681i)2-s + (0.408 − 0.408i)3-s + (0.0722 − 0.997i)4-s + (0.0576 + 0.0576i)5-s + (0.0208 − 0.576i)6-s + 0.385i·7-s + (−0.626 − 0.779i)8-s − 0.333i·9-s + (0.0815 + 0.00294i)10-s + (0.697 + 0.697i)11-s + (−0.377 − 0.436i)12-s + (−0.311 + 0.311i)13-s + (0.262 + 0.282i)14-s + 0.0470·15-s + (−0.989 − 0.144i)16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.66743 - 1.29773i\)
\(L(\frac12)\) \(\approx\) \(1.66743 - 1.29773i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.07 + 1.92i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (-0.644 - 0.644i)T + 125iT^{2} \)
7 \( 1 - 7.13iT - 343T^{2} \)
11 \( 1 + (-25.4 - 25.4i)T + 1.33e3iT^{2} \)
13 \( 1 + (14.6 - 14.6i)T - 2.19e3iT^{2} \)
17 \( 1 - 71.4T + 4.91e3T^{2} \)
19 \( 1 + (43.6 - 43.6i)T - 6.85e3iT^{2} \)
23 \( 1 - 211. iT - 1.21e4T^{2} \)
29 \( 1 + (-5.84 + 5.84i)T - 2.43e4iT^{2} \)
31 \( 1 + 107.T + 2.97e4T^{2} \)
37 \( 1 + (184. + 184. i)T + 5.06e4iT^{2} \)
41 \( 1 + 360. iT - 6.89e4T^{2} \)
43 \( 1 + (312. + 312. i)T + 7.95e4iT^{2} \)
47 \( 1 - 343.T + 1.03e5T^{2} \)
53 \( 1 + (249. + 249. i)T + 1.48e5iT^{2} \)
59 \( 1 + (152. + 152. i)T + 2.05e5iT^{2} \)
61 \( 1 + (525. - 525. i)T - 2.26e5iT^{2} \)
67 \( 1 + (35.3 - 35.3i)T - 3.00e5iT^{2} \)
71 \( 1 - 784. iT - 3.57e5T^{2} \)
73 \( 1 - 800. iT - 3.89e5T^{2} \)
79 \( 1 - 548.T + 4.93e5T^{2} \)
83 \( 1 + (-464. + 464. i)T - 5.71e5iT^{2} \)
89 \( 1 + 302. iT - 7.04e5T^{2} \)
97 \( 1 - 1.56e3T + 9.12e5T^{2} \)
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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

−14.59307221798689042800000539930, −13.82054305269399844600570273314, −12.45005522277921246238930300170, −11.85523159423535311391442761002, −10.21354236184952154293016442931, −9.084110356338710462663964946918, −7.17652970955881941757735128490, −5.62133456576619684215319282490, −3.76383326348590999181676295764, −1.89639793809457506293580991513, 3.26537233622604269126769156378, 4.76926747303338639058689496557, 6.39366568277001339694660858293, 7.88232635379356918272831380512, 9.085028832409410485228935633770, 10.75927436158750016241833926139, 12.20766416791951742614293104557, 13.40821222087276057948365305342, 14.40717708505037761413901070900, 15.15466151548988685839245729535

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