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Properties

Label 2-384-48.5-c2-0-20
Degree $2$
Conductor $384$
Sign $-0.204 + 0.978i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
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  + (−2.77 − 1.13i)3-s + (6.28 − 6.28i)5-s − 1.64i·7-s + (6.43 + 6.29i)9-s + (4.75 − 4.75i)11-s + (9.35 − 9.35i)13-s + (−24.5 + 10.3i)15-s + 11.4i·17-s + (−8.58 + 8.58i)19-s + (−1.86 + 4.57i)21-s + 16.2·23-s − 54.0i·25-s + (−10.7 − 24.7i)27-s + (−10.7 − 10.7i)29-s − 6.35·31-s + ⋯
L(s)  = 1  + (−0.926 − 0.377i)3-s + (1.25 − 1.25i)5-s − 0.235i·7-s + (0.715 + 0.699i)9-s + (0.432 − 0.432i)11-s + (0.719 − 0.719i)13-s + (−1.63 + 0.689i)15-s + 0.675i·17-s + (−0.451 + 0.451i)19-s + (−0.0887 + 0.217i)21-s + 0.706·23-s − 2.16i·25-s + (−0.398 − 0.917i)27-s + (−0.370 − 0.370i)29-s − 0.204·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.970590 - 1.19480i\)
\(L(\frac12)\) \(\approx\) \(0.970590 - 1.19480i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.77 + 1.13i)T \)
good5 \( 1 + (-6.28 + 6.28i)T - 25iT^{2} \)
7 \( 1 + 1.64iT - 49T^{2} \)
11 \( 1 + (-4.75 + 4.75i)T - 121iT^{2} \)
13 \( 1 + (-9.35 + 9.35i)T - 169iT^{2} \)
17 \( 1 - 11.4iT - 289T^{2} \)
19 \( 1 + (8.58 - 8.58i)T - 361iT^{2} \)
23 \( 1 - 16.2T + 529T^{2} \)
29 \( 1 + (10.7 + 10.7i)T + 841iT^{2} \)
31 \( 1 + 6.35T + 961T^{2} \)
37 \( 1 + (27.2 + 27.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.98T + 1.68e3T^{2} \)
43 \( 1 + (19.4 + 19.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 74.9iT - 2.20e3T^{2} \)
53 \( 1 + (-4.00 + 4.00i)T - 2.80e3iT^{2} \)
59 \( 1 + (-27.9 + 27.9i)T - 3.48e3iT^{2} \)
61 \( 1 + (39.2 - 39.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (68.6 - 68.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 40.6T + 5.04e3T^{2} \)
73 \( 1 - 59.0iT - 5.32e3T^{2} \)
79 \( 1 + 17.3T + 6.24e3T^{2} \)
83 \( 1 + (-75.1 - 75.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 78.8T + 7.92e3T^{2} \)
97 \( 1 + 38.8T + 9.40e3T^{2} \)
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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

−10.74691411628915490332673603075, −10.13352145749883686183565810860, −8.987312386785573521929471681391, −8.231848607431426161728138185144, −6.80287054489358368335173029471, −5.79477743895559087447348902181, −5.36913533146669177007382090694, −4.03489805672819057923152171906, −1.85049573281990042813794351540, −0.820575488530630287152921197145, 1.68346271949407370012550905130, 3.15409435807700063500726740430, 4.63231053605361000194557309691, 5.77507256203536068333959101062, 6.54858708917713102962573895363, 7.12684692517980016442731945226, 9.084438170858864940302327615185, 9.626767644918521125330189236189, 10.64269310480593204295431121969, 11.11574010324981280145628713767

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