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Properties

Label 2-384-48.5-c2-0-12
Degree $2$
Conductor $384$
Sign $-0.249 - 0.968i$
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

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19597 + 1.54279i\)
\(L(\frac12)\) \(\approx\) \(1.19597 + 1.54279i\)
\(L(2)\) 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

−11.35718322080001391400198221058, −10.43276803591504756472420101247, −9.449548449046982934792845323395, −8.511420297979630433019559747507, −8.068191038628073027632768613144, −6.61814739248475699583447412784, −5.64163620660273569741039917792, −4.06769285920214802809779897634, −3.35207013651714682988300587196, −2.01049445344633146170850902611, 0.802539649357753360204362816719, 2.27173218779678413416540058141, 3.99894793062633627187472467555, 4.40776100463164273652950047397, 6.42991889451752569082975238791, 7.21776397726315551092397209148, 7.980425978869912446246982385631, 9.014231919680626299475384825067, 9.692500141697563438776759697046, 10.87655512769640493580452666170

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