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Properties

Label 2-48-16.11-c6-0-12
Degree $2$
Conductor $48$
Sign $-0.787 - 0.616i$
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  + (1.02 + 7.93i)2-s + (11.0 + 11.0i)3-s + (−61.9 + 16.2i)4-s + (145. + 145. i)5-s + (−76.1 + 98.7i)6-s + 535.·7-s + (−192. − 474. i)8-s + 242. i·9-s + (−1.00e3 + 1.30e3i)10-s + (−1.23e3 + 1.23e3i)11-s + (−861. − 503. i)12-s + (1.76e3 − 1.76e3i)13-s + (548. + 4.25e3i)14-s + 3.20e3i·15-s + (3.56e3 − 2.01e3i)16-s − 5.53e3·17-s + ⋯
L(s)  = 1  + (0.128 + 0.991i)2-s + (0.408 + 0.408i)3-s + (−0.967 + 0.253i)4-s + (1.16 + 1.16i)5-s + (−0.352 + 0.457i)6-s + 1.56·7-s + (−0.375 − 0.926i)8-s + 0.333i·9-s + (−1.00 + 1.30i)10-s + (−0.928 + 0.928i)11-s + (−0.498 − 0.291i)12-s + (0.803 − 0.803i)13-s + (0.200 + 1.54i)14-s + 0.950i·15-s + (0.871 − 0.491i)16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.787 - 0.616i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.809550 + 2.34584i\)
\(L(\frac12)\) \(\approx\) \(0.809550 + 2.34584i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 - 7.93i)T \)
3 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (-145. - 145. i)T + 1.56e4iT^{2} \)
7 \( 1 - 535.T + 1.17e5T^{2} \)
11 \( 1 + (1.23e3 - 1.23e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (-1.76e3 + 1.76e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 5.53e3T + 2.41e7T^{2} \)
19 \( 1 + (3.22e3 + 3.22e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 2.83e3T + 1.48e8T^{2} \)
29 \( 1 + (-1.17e4 + 1.17e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 3.11e4iT - 8.87e8T^{2} \)
37 \( 1 + (-3.00e4 - 3.00e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 2.42e4iT - 4.75e9T^{2} \)
43 \( 1 + (-3.09e4 + 3.09e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 6.26e3iT - 1.07e10T^{2} \)
53 \( 1 + (-1.98e5 - 1.98e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-1.34e5 + 1.34e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.03e5 + 1.03e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (3.96e5 + 3.96e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 3.34e5T + 1.28e11T^{2} \)
73 \( 1 - 4.36e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.50e5iT - 2.43e11T^{2} \)
83 \( 1 + (4.60e5 + 4.60e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.37e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.47e5T + 8.32e11T^{2} \)
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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

−15.03769774572945171790960132009, −13.95455020877431673268614315568, −13.21330694862264325969623483161, −10.95854718282312248369180248522, −10.01243749560410664913658028535, −8.532572432981155483181831921152, −7.37862722812966549275390470378, −5.87603176028270267959276470377, −4.56293223340313523322699270363, −2.36243354627509818579040571335, 1.18173953646584308466761814384, 2.15121763670586441215231314041, 4.51331004499073967506875889169, 5.70664634045918520779006277264, 8.445983162778018390959286074189, 8.827715720739000711775747878937, 10.51160171778803182393896646509, 11.62055288688513443290731065557, 13.01072966973264156876409455247, 13.63637739479481768489926858969

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