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Properties

Label 2-384-48.5-c2-0-9
Degree $2$
Conductor $384$
Sign $0.968 - 0.249i$
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  + (−2.59 − 1.50i)3-s + (−2.59 + 2.59i)5-s − 7.30i·7-s + (4.47 + 7.81i)9-s + (−11.3 + 11.3i)11-s + (0.746 − 0.746i)13-s + (10.6 − 2.83i)15-s + 6.67i·17-s + (22.1 − 22.1i)19-s + (−10.9 + 18.9i)21-s + 21.4·23-s + 11.4i·25-s + (0.153 − 26.9i)27-s + (1.54 + 1.54i)29-s + 14.6·31-s + ⋯
L(s)  = 1  + (−0.865 − 0.501i)3-s + (−0.519 + 0.519i)5-s − 1.04i·7-s + (0.496 + 0.867i)9-s + (−1.02 + 1.02i)11-s + (0.0574 − 0.0574i)13-s + (0.710 − 0.188i)15-s + 0.392i·17-s + (1.16 − 1.16i)19-s + (−0.523 + 0.902i)21-s + 0.932·23-s + 0.459i·25-s + (0.00567 − 0.999i)27-s + (0.0531 + 0.0531i)29-s + 0.471·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.968 - 0.249i$
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: \((2,\ 384,\ (\ :1),\ 0.968 - 0.249i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.983819 + 0.124583i\)
\(L(\frac12)\) \(\approx\) \(0.983819 + 0.124583i\)
\(L(2)\) 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 \( 1 + (2.59 + 1.50i)T \)
good5 \( 1 + (2.59 - 2.59i)T - 25iT^{2} \)
7 \( 1 + 7.30iT - 49T^{2} \)
11 \( 1 + (11.3 - 11.3i)T - 121iT^{2} \)
13 \( 1 + (-0.746 + 0.746i)T - 169iT^{2} \)
17 \( 1 - 6.67iT - 289T^{2} \)
19 \( 1 + (-22.1 + 22.1i)T - 361iT^{2} \)
23 \( 1 - 21.4T + 529T^{2} \)
29 \( 1 + (-1.54 - 1.54i)T + 841iT^{2} \)
31 \( 1 - 14.6T + 961T^{2} \)
37 \( 1 + (-50.1 - 50.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 15.0T + 1.68e3T^{2} \)
43 \( 1 + (-26.3 - 26.3i)T + 1.84e3iT^{2} \)
47 \( 1 - 36.6iT - 2.20e3T^{2} \)
53 \( 1 + (-50.9 + 50.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (-12.1 + 12.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-27.5 + 27.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (4.84 - 4.84i)T - 4.48e3iT^{2} \)
71 \( 1 + 74.9T + 5.04e3T^{2} \)
73 \( 1 + 3.47iT - 5.32e3T^{2} \)
79 \( 1 + 103.T + 6.24e3T^{2} \)
83 \( 1 + (-31.7 - 31.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 78.2T + 7.92e3T^{2} \)
97 \( 1 + 61.5T + 9.40e3T^{2} \)
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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

−11.13589338384513236134015844418, −10.50245109375308305314913268441, −9.619186081727078870777096450011, −7.892395884977301634638700440735, −7.33726937712598065325307976852, −6.66857042814489039491298426011, −5.23862396053260267204761313388, −4.38235530154973881274699158279, −2.80215535493993949452762275944, −0.962872146062409076762713723734, 0.67970958113311047006855299424, 2.89579713722635360191509203109, 4.23822617953702751274618553098, 5.48746436459730761422867132022, 5.79219550632887294607315086370, 7.37192219874334287567317230817, 8.434378095294910085332649883042, 9.248731090030229847426797076683, 10.28805471239380926983371805286, 11.19870914641707395020544864649

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