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Properties

Label 2-384-48.5-c2-0-5
Degree $2$
Conductor $384$
Sign $-0.274 - 0.961i$
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  + (2.17 − 2.06i)3-s + (−3.17 + 3.17i)5-s + 6.03i·7-s + (0.485 − 8.98i)9-s + (−13.0 + 13.0i)11-s + (−6.39 + 6.39i)13-s + (−0.363 + 13.4i)15-s + 4.39i·17-s + (3.21 − 3.21i)19-s + (12.4 + 13.1i)21-s − 34.0·23-s + 4.78i·25-s + (−17.4 − 20.5i)27-s + (27.9 + 27.9i)29-s − 7.90·31-s + ⋯
L(s)  = 1  + (0.725 − 0.687i)3-s + (−0.635 + 0.635i)5-s + 0.862i·7-s + (0.0539 − 0.998i)9-s + (−1.18 + 1.18i)11-s + (−0.491 + 0.491i)13-s + (−0.0242 + 0.898i)15-s + 0.258i·17-s + (0.168 − 0.168i)19-s + (0.593 + 0.626i)21-s − 1.47·23-s + 0.191i·25-s + (−0.647 − 0.761i)27-s + (0.964 + 0.964i)29-s − 0.255·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.274 - 0.961i$
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: \((2,\ 384,\ (\ :1),\ -0.274 - 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.664037 + 0.880139i\)
\(L(\frac12)\) \(\approx\) \(0.664037 + 0.880139i\)
\(L(2)\) 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 \( 1 + (-2.17 + 2.06i)T \)
good5 \( 1 + (3.17 - 3.17i)T - 25iT^{2} \)
7 \( 1 - 6.03iT - 49T^{2} \)
11 \( 1 + (13.0 - 13.0i)T - 121iT^{2} \)
13 \( 1 + (6.39 - 6.39i)T - 169iT^{2} \)
17 \( 1 - 4.39iT - 289T^{2} \)
19 \( 1 + (-3.21 + 3.21i)T - 361iT^{2} \)
23 \( 1 + 34.0T + 529T^{2} \)
29 \( 1 + (-27.9 - 27.9i)T + 841iT^{2} \)
31 \( 1 + 7.90T + 961T^{2} \)
37 \( 1 + (20.0 + 20.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 45.1T + 1.68e3T^{2} \)
43 \( 1 + (-36.0 - 36.0i)T + 1.84e3iT^{2} \)
47 \( 1 + 5.08iT - 2.20e3T^{2} \)
53 \( 1 + (-20.7 + 20.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (39.0 - 39.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-49.8 + 49.8i)T - 3.72e3iT^{2} \)
67 \( 1 + (44.9 - 44.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 46.6T + 5.04e3T^{2} \)
73 \( 1 - 97.3iT - 5.32e3T^{2} \)
79 \( 1 + 40.1T + 6.24e3T^{2} \)
83 \( 1 + (35.5 + 35.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 69.6T + 7.92e3T^{2} \)
97 \( 1 - 61.0T + 9.40e3T^{2} \)
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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

−11.62212158317460070566640854581, −10.40108360827325368637125327791, −9.486142747058272274362178045458, −8.471842844610777663989788410000, −7.56758018373411159188483578439, −7.01517170665722758581017148934, −5.72278868590466788597939720461, −4.30199506629665709918941377610, −2.90487998424246838809065856788, −2.05557486707129818188285189151, 0.42102153555175260934399637179, 2.65446769704493821629830630552, 3.82137417858184122766933607588, 4.68800693792824858188491554327, 5.79951885081004383572364303181, 7.61978345045776287341134409025, 8.006751642583810885366319374821, 8.882947232849111876166631771878, 10.14829392578368226267085308923, 10.53854311519583750669275218166

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