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Properties

Label 2-384-48.29-c2-0-17
Degree $2$
Conductor $384$
Sign $0.577 + 0.816i$
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  + (−1.18 + 2.75i)3-s + (0.00985 + 0.00985i)5-s − 6.42i·7-s + (−6.19 − 6.53i)9-s + (9.07 + 9.07i)11-s + (−12.6 − 12.6i)13-s + (−0.0388 + 0.0154i)15-s − 19.0i·17-s + (−2.07 − 2.07i)19-s + (17.7 + 7.61i)21-s − 19.5·23-s − 24.9i·25-s + (25.3 − 9.32i)27-s + (11.1 − 11.1i)29-s + 59.9·31-s + ⋯
L(s)  = 1  + (−0.395 + 0.918i)3-s + (0.00197 + 0.00197i)5-s − 0.917i·7-s + (−0.687 − 0.725i)9-s + (0.824 + 0.824i)11-s + (−0.969 − 0.969i)13-s + (−0.00259 + 0.00103i)15-s − 1.11i·17-s + (−0.109 − 0.109i)19-s + (0.842 + 0.362i)21-s − 0.850·23-s − 0.999i·25-s + (0.938 − 0.345i)27-s + (0.385 − 0.385i)29-s + 1.93·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.981265 - 0.508105i\)
\(L(\frac12)\) \(\approx\) \(0.981265 - 0.508105i\)
\(L(2)\) 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 \( 1 + (1.18 - 2.75i)T \)
good5 \( 1 + (-0.00985 - 0.00985i)T + 25iT^{2} \)
7 \( 1 + 6.42iT - 49T^{2} \)
11 \( 1 + (-9.07 - 9.07i)T + 121iT^{2} \)
13 \( 1 + (12.6 + 12.6i)T + 169iT^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (2.07 + 2.07i)T + 361iT^{2} \)
23 \( 1 + 19.5T + 529T^{2} \)
29 \( 1 + (-11.1 + 11.1i)T - 841iT^{2} \)
31 \( 1 - 59.9T + 961T^{2} \)
37 \( 1 + (9.32 - 9.32i)T - 1.36e3iT^{2} \)
41 \( 1 - 47.2T + 1.68e3T^{2} \)
43 \( 1 + (-24.1 + 24.1i)T - 1.84e3iT^{2} \)
47 \( 1 - 6.29iT - 2.20e3T^{2} \)
53 \( 1 + (20.6 + 20.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (60.3 + 60.3i)T + 3.48e3iT^{2} \)
61 \( 1 + (48.0 + 48.0i)T + 3.72e3iT^{2} \)
67 \( 1 + (23.7 + 23.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 13.5T + 5.04e3T^{2} \)
73 \( 1 - 31.4iT - 5.32e3T^{2} \)
79 \( 1 + 47.4T + 6.24e3T^{2} \)
83 \( 1 + (70.3 - 70.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 95.1T + 7.92e3T^{2} \)
97 \( 1 - 61.6T + 9.40e3T^{2} \)
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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

−10.81695581336087573422264706946, −9.956134555969956380527018793911, −9.604916931772454022012141344128, −8.207487660807850352371050991401, −7.15725366004801434938852224979, −6.14820481142849072022362907814, −4.79730660846037572202828378476, −4.22633287772968513621241790816, −2.79180750832344804208462705653, −0.53757781143296179816489910721, 1.46641592552183654953392966279, 2.72195603313904103465738550521, 4.40203811293161631949592111636, 5.81247240808412952359591472491, 6.33027439488424401961286242606, 7.46463248436064743382712050726, 8.493960609544206362674767922285, 9.220628517035075231983097337665, 10.51057377911119748427189563538, 11.61598483340559198910914275712

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