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Properties

Label 2-384-12.11-c5-0-26
Degree $2$
Conductor $384$
Sign $-0.791 - 0.610i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.3 + 9.51i)3-s + 26.7i·5-s + 70.6i·7-s + (61.8 + 235. i)9-s + 98.7·11-s + 210.·13-s + (−254. + 330. i)15-s + 524. i·17-s + 1.11e3i·19-s + (−672. + 872. i)21-s + 866.·23-s + 2.40e3·25-s + (−1.47e3 + 3.48e3i)27-s − 2.16e3i·29-s + 1.72e3i·31-s + ⋯
L(s)  = 1  + (0.791 + 0.610i)3-s + 0.478i·5-s + 0.545i·7-s + (0.254 + 0.967i)9-s + 0.246·11-s + 0.345·13-s + (−0.292 + 0.379i)15-s + 0.440i·17-s + 0.708i·19-s + (−0.332 + 0.431i)21-s + 0.341·23-s + 0.770·25-s + (−0.388 + 0.921i)27-s − 0.477i·29-s + 0.322i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.791 - 0.610i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.791 - 0.610i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.481311397\)
\(L(\frac12)\) \(\approx\) \(2.481311397\)
\(L(\frac{7}{2})\) 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 \( 1 + (-12.3 - 9.51i)T \)
good5 \( 1 - 26.7iT - 3.12e3T^{2} \)
7 \( 1 - 70.6iT - 1.68e4T^{2} \)
11 \( 1 - 98.7T + 1.61e5T^{2} \)
13 \( 1 - 210.T + 3.71e5T^{2} \)
17 \( 1 - 524. iT - 1.41e6T^{2} \)
19 \( 1 - 1.11e3iT - 2.47e6T^{2} \)
23 \( 1 - 866.T + 6.43e6T^{2} \)
29 \( 1 + 2.16e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.72e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.23e4T + 6.93e7T^{2} \)
41 \( 1 - 1.39e3iT - 1.15e8T^{2} \)
43 \( 1 - 9.13e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.18e4T + 2.29e8T^{2} \)
53 \( 1 - 7.60e3iT - 4.18e8T^{2} \)
59 \( 1 - 4.09e4T + 7.14e8T^{2} \)
61 \( 1 + 2.41e4T + 8.44e8T^{2} \)
67 \( 1 - 8.96e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.75e3T + 1.80e9T^{2} \)
73 \( 1 + 6.50e4T + 2.07e9T^{2} \)
79 \( 1 + 1.00e5iT - 3.07e9T^{2} \)
83 \( 1 + 2.90e4T + 3.93e9T^{2} \)
89 \( 1 + 619. iT - 5.58e9T^{2} \)
97 \( 1 + 7.49e4T + 8.58e9T^{2} \)
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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

−10.67105157923892812507511103656, −10.00057555020082671543352678477, −8.954836229706897124168491289228, −8.342318084437922867083677496392, −7.24221716593473691763062262061, −6.09555768721931728927250270119, −4.92586473700163501556448115259, −3.73975545644484795585582518006, −2.85408598669585016059565162478, −1.62993914142993001569670614140, 0.54431613956459503222109376556, 1.56674260349666092245873533560, 2.93297562962182305549516759355, 4.00951711941724757387585233175, 5.22033262541475945942696039992, 6.68193551861444475857354418463, 7.26825379242690903455961648888, 8.450646881164754777042165037422, 8.994954528565479425587421027523, 10.00738310783369871998403628539

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