Properties

Label 2-384-48.5-c2-0-8
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: $\chi_{384} (353, \cdot )$
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

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

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