Properties

Label 2-384-24.11-c5-0-20
Degree $2$
Conductor $384$
Sign $-0.789 - 0.613i$
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  + (−15.4 + 1.94i)3-s − 101.·5-s + 200. i·7-s + (235. − 60.0i)9-s + 350. i·11-s − 744. i·13-s + (1.56e3 − 196. i)15-s + 1.44e3i·17-s + 1.31e3·19-s + (−389. − 3.09e3i)21-s + 2.04e3·23-s + 7.09e3·25-s + (−3.52e3 + 1.38e3i)27-s + 3.78e3·29-s − 1.42e3i·31-s + ⋯
L(s)  = 1  + (−0.992 + 0.124i)3-s − 1.80·5-s + 1.54i·7-s + (0.968 − 0.247i)9-s + 0.872i·11-s − 1.22i·13-s + (1.79 − 0.225i)15-s + 1.21i·17-s + 0.834·19-s + (−0.192 − 1.53i)21-s + 0.804·23-s + 2.27·25-s + (−0.930 + 0.366i)27-s + 0.835·29-s − 0.266i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.789 - 0.613i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8024797490\)
\(L(\frac12)\) \(\approx\) \(0.8024797490\)
\(L(\frac{7}{2})\) 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 \( 1 + (15.4 - 1.94i)T \)
good5 \( 1 + 101.T + 3.12e3T^{2} \)
7 \( 1 - 200. iT - 1.68e4T^{2} \)
11 \( 1 - 350. iT - 1.61e5T^{2} \)
13 \( 1 + 744. iT - 3.71e5T^{2} \)
17 \( 1 - 1.44e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.31e3T + 2.47e6T^{2} \)
23 \( 1 - 2.04e3T + 6.43e6T^{2} \)
29 \( 1 - 3.78e3T + 2.05e7T^{2} \)
31 \( 1 + 1.42e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.61e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.08e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.61e4T + 1.47e8T^{2} \)
47 \( 1 - 3.41e3T + 2.29e8T^{2} \)
53 \( 1 - 539.T + 4.18e8T^{2} \)
59 \( 1 - 3.37e4iT - 7.14e8T^{2} \)
61 \( 1 - 5.04e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.57e4T + 1.35e9T^{2} \)
71 \( 1 - 3.64e4T + 1.80e9T^{2} \)
73 \( 1 + 3.70e4T + 2.07e9T^{2} \)
79 \( 1 + 3.32e4iT - 3.07e9T^{2} \)
83 \( 1 - 7.58e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.73e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.15e5T + 8.58e9T^{2} \)
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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

−11.08748910162305268207580334314, −10.18986921887159645126651957813, −8.943930837839687738652863829895, −8.000779098431943799156427774811, −7.21449544152026522697665462497, −5.96525310823983340024157257175, −5.06737704308579709459868689955, −4.09553105243087262428067340196, −2.85565833623841989502627638411, −0.969131108652346914174487411616, 0.39692012704761600418159544907, 0.932940830658210810541701664417, 3.38083957589660557209195093021, 4.24676700063401594430036732515, 4.99407473697915635398408756782, 6.72839763467754447605869776629, 7.19116769085818998294410140142, 7.962705440704539132253026853989, 9.283681949613612720719322575755, 10.57620813649173511902246447238

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