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Properties

Label 2-384-16.11-c6-0-24
Degree $2$
Conductor $384$
Sign $-0.180 + 0.983i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.0 − 11.0i)3-s + (−128. − 128. i)5-s − 76.4·7-s + 242. i·9-s + (−565. + 565. i)11-s + (−2.77e3 + 2.77e3i)13-s + 2.82e3i·15-s + 5.24e3·17-s + (−46.8 − 46.8i)19-s + (842. + 842. i)21-s + 1.95e4·23-s + 1.72e4i·25-s + (2.67e3 − 2.67e3i)27-s + (1.64e4 − 1.64e4i)29-s − 2.40e4i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−1.02 − 1.02i)5-s − 0.222·7-s + 0.333i·9-s + (−0.425 + 0.425i)11-s + (−1.26 + 1.26i)13-s + 0.838i·15-s + 1.06·17-s + (−0.00682 − 0.00682i)19-s + (0.0909 + 0.0909i)21-s + 1.60·23-s + 1.10i·25-s + (0.136 − 0.136i)27-s + (0.674 − 0.674i)29-s − 0.808i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.180 + 0.983i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ -0.180 + 0.983i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.8023518663\)
\(L(\frac12)\) \(\approx\) \(0.8023518663\)
\(L(4)\) 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 \( 1 + (11.0 + 11.0i)T \)
good5 \( 1 + (128. + 128. i)T + 1.56e4iT^{2} \)
7 \( 1 + 76.4T + 1.17e5T^{2} \)
11 \( 1 + (565. - 565. i)T - 1.77e6iT^{2} \)
13 \( 1 + (2.77e3 - 2.77e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 5.24e3T + 2.41e7T^{2} \)
19 \( 1 + (46.8 + 46.8i)T + 4.70e7iT^{2} \)
23 \( 1 - 1.95e4T + 1.48e8T^{2} \)
29 \( 1 + (-1.64e4 + 1.64e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 2.40e4iT - 8.87e8T^{2} \)
37 \( 1 + (-3.81e4 - 3.81e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 6.65e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.49e4 - 2.49e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 1.20e5iT - 1.07e10T^{2} \)
53 \( 1 + (9.27e4 + 9.27e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (1.60e5 - 1.60e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-1.78e5 + 1.78e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (2.74e5 + 2.74e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 6.31e5T + 1.28e11T^{2} \)
73 \( 1 + 4.70e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.35e5iT - 2.43e11T^{2} \)
83 \( 1 + (2.84e5 + 2.84e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 2.72e5iT - 4.96e11T^{2} \)
97 \( 1 - 4.99e5T + 8.32e11T^{2} \)
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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

−9.946088850292300289545043176199, −9.193953100439378217512236841618, −7.985490494550200048011027368224, −7.44618361455407179836809939288, −6.34640966711925064065092162050, −4.89189673019029595537447870428, −4.52204911488415297641432705837, −2.92044870121634358382420746295, −1.41815363592905686780080850173, −0.31556391742711539636016382385, 0.68086333243654919295594453768, 2.91434183671481021568055774364, 3.34416138806872786836567447847, 4.81082473706984786255131738590, 5.69055269165269029029761574621, 7.05756796016258812288676215061, 7.56067444806804329389516572467, 8.691296022168612388049367265484, 10.06436723789121434779995459427, 10.53663097160318885711127551193

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