Properties

Label 2-48-16.3-c6-0-0
Degree $2$
Conductor $48$
Sign $-0.736 + 0.676i$
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  + (5.00 + 6.23i)2-s + (−11.0 + 11.0i)3-s + (−13.8 + 62.4i)4-s + (45.7 − 45.7i)5-s + (−123. − 13.5i)6-s − 565.·7-s + (−458. + 226. i)8-s − 242. i·9-s + (514. + 56.1i)10-s + (1.29e3 + 1.29e3i)11-s + (−536. − 841. i)12-s + (−2.88e3 − 2.88e3i)13-s + (−2.83e3 − 3.52e3i)14-s + 1.00e3i·15-s + (−3.71e3 − 1.72e3i)16-s − 662.·17-s + ⋯
L(s)  = 1  + (0.626 + 0.779i)2-s + (−0.408 + 0.408i)3-s + (−0.215 + 0.976i)4-s + (0.366 − 0.366i)5-s + (−0.573 − 0.0626i)6-s − 1.64·7-s + (−0.896 + 0.443i)8-s − 0.333i·9-s + (0.514 + 0.0561i)10-s + (0.970 + 0.970i)11-s + (−0.310 − 0.486i)12-s + (−1.31 − 1.31i)13-s + (−1.03 − 1.28i)14-s + 0.299i·15-s + (−0.906 − 0.421i)16-s − 0.134·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.736 + 0.676i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.221445 - 0.568215i\)
\(L(\frac12)\) \(\approx\) \(0.221445 - 0.568215i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.00 - 6.23i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (-45.7 + 45.7i)T - 1.56e4iT^{2} \)
7 \( 1 + 565.T + 1.17e5T^{2} \)
11 \( 1 + (-1.29e3 - 1.29e3i)T + 1.77e6iT^{2} \)
13 \( 1 + (2.88e3 + 2.88e3i)T + 4.82e6iT^{2} \)
17 \( 1 + 662.T + 2.41e7T^{2} \)
19 \( 1 + (-1.94e3 + 1.94e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.58e4T + 1.48e8T^{2} \)
29 \( 1 + (-1.15e4 - 1.15e4i)T + 5.94e8iT^{2} \)
31 \( 1 - 3.90e4iT - 8.87e8T^{2} \)
37 \( 1 + (5.13e4 - 5.13e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 6.65e4iT - 4.75e9T^{2} \)
43 \( 1 + (-6.89e4 - 6.89e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.07e5iT - 1.07e10T^{2} \)
53 \( 1 + (1.08e5 - 1.08e5i)T - 2.21e10iT^{2} \)
59 \( 1 + (6.78e4 + 6.78e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (9.92e4 + 9.92e4i)T + 5.15e10iT^{2} \)
67 \( 1 + (1.05e5 - 1.05e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 2.66e5T + 1.28e11T^{2} \)
73 \( 1 + 2.67e5iT - 1.51e11T^{2} \)
79 \( 1 + 5.57e5iT - 2.43e11T^{2} \)
83 \( 1 + (2.36e5 - 2.36e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 4.19e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.93e5T + 8.32e11T^{2} \)
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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

−15.31543664126766974634499075426, −14.05953870163335134148838032349, −12.62402509510520481148425948339, −12.29876584707835994675276951738, −10.05852785591368713867787683324, −9.204020468000351911455609325790, −7.24614899316044648147606730820, −6.12746822980372079863285789165, −4.82877984041839360554075142181, −3.21571315000693033041433674531, 0.22077824399636692929110079186, 2.31665537858873895197827886251, 3.92261942234703074213802808388, 5.98190454098295953522679813624, 6.70868182793713874390853418268, 9.323176084512378573588766335751, 10.13882924466567058679458246178, 11.66884901225104666528682092472, 12.37324205236260737748423478470, 13.66537675722227090979665208047

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