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Properties

Label 2-384-16.11-c6-0-31
Degree $2$
Conductor $384$
Sign $-0.924 + 0.382i$
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 + (−29.4 − 29.4i)5-s − 461.·7-s + 242. i·9-s + (−1.18e3 + 1.18e3i)11-s + (2.43e3 − 2.43e3i)13-s + 650. i·15-s + 4.81e3·17-s + (8.31e3 + 8.31e3i)19-s + (5.08e3 + 5.08e3i)21-s + 8.17e3·23-s − 1.38e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−1.67e4 + 1.67e4i)29-s − 7.90e3i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.235 − 0.235i)5-s − 1.34·7-s + 0.333i·9-s + (−0.891 + 0.891i)11-s + (1.10 − 1.10i)13-s + 0.192i·15-s + 0.980·17-s + (1.21 + 1.21i)19-s + (0.548 + 0.548i)21-s + 0.672·23-s − 0.888i·25-s + (0.136 − 0.136i)27-s + (−0.688 + 0.688i)29-s − 0.265i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.382i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.924 + 0.382i$
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: \((2,\ 384,\ (\ :3),\ -0.924 + 0.382i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.5249435234\)
\(L(\frac12)\) \(\approx\) \(0.5249435234\)
\(L(4)\) 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 \( 1 + (29.4 + 29.4i)T + 1.56e4iT^{2} \)
7 \( 1 + 461.T + 1.17e5T^{2} \)
11 \( 1 + (1.18e3 - 1.18e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (-2.43e3 + 2.43e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 4.81e3T + 2.41e7T^{2} \)
19 \( 1 + (-8.31e3 - 8.31e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 8.17e3T + 1.48e8T^{2} \)
29 \( 1 + (1.67e4 - 1.67e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 7.90e3iT - 8.87e8T^{2} \)
37 \( 1 + (2.21e4 + 2.21e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 7.14e4iT - 4.75e9T^{2} \)
43 \( 1 + (-7.07e4 + 7.07e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 3.15e3iT - 1.07e10T^{2} \)
53 \( 1 + (-2.04e3 - 2.04e3i)T + 2.21e10iT^{2} \)
59 \( 1 + (2.55e5 - 2.55e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (2.86e5 - 2.86e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (-2.92e5 - 2.92e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 1.28e5T + 1.28e11T^{2} \)
73 \( 1 + 7.98e4iT - 1.51e11T^{2} \)
79 \( 1 + 1.38e5iT - 2.43e11T^{2} \)
83 \( 1 + (1.36e5 + 1.36e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 - 4.69e4T + 8.32e11T^{2} \)
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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

−10.17317474654588383814442667059, −9.054389873084432442785312137989, −7.83413685773824433539931865656, −7.23182742561194659274748999006, −5.92218782838157256949237979891, −5.39861980975522252064413854528, −3.77791131325047904206231387928, −2.86261780617512935161941589813, −1.23544731659255668699993999149, −0.15872968405990706183476199790, 1.01488494191178072141738194098, 3.03318742215601789392190488493, 3.53720305355493435516565619682, 5.01549467410210698997513962892, 6.01608047857979860998939544715, 6.79694360236475899198424157013, 7.914042829421143897302781009976, 9.248159716506148409175227538634, 9.642218709070251165589767908646, 11.00461973689464998697305941894

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