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Properties

Label 2-48-16.3-c6-0-17
Degree $2$
Conductor $48$
Sign $-0.379 + 0.925i$
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  + (5.65 − 5.66i)2-s + (−11.0 + 11.0i)3-s + (−0.0944 − 63.9i)4-s + (29.4 − 29.4i)5-s + (0.0919 + 124. i)6-s + 461.·7-s + (−362. − 361. i)8-s − 242. i·9-s + (−0.246 − 333. i)10-s + (−1.18e3 − 1.18e3i)11-s + (706. + 704. i)12-s + (−2.43e3 − 2.43e3i)13-s + (2.60e3 − 2.61e3i)14-s + 650. i·15-s + (−4.09e3 + 12.0i)16-s + 4.81e3·17-s + ⋯
L(s)  = 1  + (0.706 − 0.707i)2-s + (−0.408 + 0.408i)3-s + (−0.00147 − 0.999i)4-s + (0.235 − 0.235i)5-s + (0.000425 + 0.577i)6-s + 1.34·7-s + (−0.708 − 0.705i)8-s − 0.333i·9-s + (−0.000246 − 0.333i)10-s + (−0.891 − 0.891i)11-s + (0.408 + 0.407i)12-s + (−1.10 − 1.10i)13-s + (0.950 − 0.951i)14-s + 0.192i·15-s + (−0.999 + 0.00295i)16-s + 0.980·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.379 + 0.925i$
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: \((2,\ 48,\ (\ :3),\ -0.379 + 0.925i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.24272 - 1.85393i\)
\(L(\frac12)\) \(\approx\) \(1.24272 - 1.85393i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 + 5.66i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (-29.4 + 29.4i)T - 1.56e4iT^{2} \)
7 \( 1 - 461.T + 1.17e5T^{2} \)
11 \( 1 + (1.18e3 + 1.18e3i)T + 1.77e6iT^{2} \)
13 \( 1 + (2.43e3 + 2.43e3i)T + 4.82e6iT^{2} \)
17 \( 1 - 4.81e3T + 2.41e7T^{2} \)
19 \( 1 + (-8.31e3 + 8.31e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 8.17e3T + 1.48e8T^{2} \)
29 \( 1 + (-1.67e4 - 1.67e4i)T + 5.94e8iT^{2} \)
31 \( 1 + 7.90e3iT - 8.87e8T^{2} \)
37 \( 1 + (-2.21e4 + 2.21e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 7.14e4iT - 4.75e9T^{2} \)
43 \( 1 + (-7.07e4 - 7.07e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 3.15e3iT - 1.07e10T^{2} \)
53 \( 1 + (2.04e3 - 2.04e3i)T - 2.21e10iT^{2} \)
59 \( 1 + (2.55e5 + 2.55e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (-2.86e5 - 2.86e5i)T + 5.15e10iT^{2} \)
67 \( 1 + (-2.92e5 + 2.92e5i)T - 9.04e10iT^{2} \)
71 \( 1 + 1.28e5T + 1.28e11T^{2} \)
73 \( 1 - 7.98e4iT - 1.51e11T^{2} \)
79 \( 1 + 1.38e5iT - 2.43e11T^{2} \)
83 \( 1 + (1.36e5 - 1.36e5i)T - 3.26e11iT^{2} \)
89 \( 1 - 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 - 4.69e4T + 8.32e11T^{2} \)
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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

−14.01620150651676230953336932487, −12.79276797273964263291628275516, −11.59727895937883508425217596299, −10.76010038524721302819109787891, −9.614208380419712860409189819700, −7.81959548464042635812413705564, −5.49990585669396605846175326449, −4.94506464900989243895277499086, −2.92147916821975624632553436348, −0.883614992414689906619330724409, 2.15021217570337058468768372645, 4.55653876788834187068002199523, 5.61699718800487737233726944693, 7.26522613248749085640537423564, 8.008676709647210006262963853531, 10.05987031038196519082492043880, 11.79656965478750276780584120169, 12.31942228050377495144602197835, 14.05620652448073914439690980457, 14.40163002773687155049122264184

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