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Properties

Label 2-384-48.5-c2-0-26
Degree $2$
Conductor $384$
Sign $0.134 + 0.990i$
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  + (2.77 − 1.14i)3-s + (4.80 − 4.80i)5-s − 7.36i·7-s + (6.35 − 6.37i)9-s + (0.514 − 0.514i)11-s + (−7.12 + 7.12i)13-s + (7.79 − 18.8i)15-s − 11.1i·17-s + (−21.1 + 21.1i)19-s + (−8.46 − 20.4i)21-s − 7.80·23-s − 21.1i·25-s + (10.2 − 24.9i)27-s + (34.6 + 34.6i)29-s − 24.8·31-s + ⋯
L(s)  = 1  + (0.923 − 0.383i)3-s + (0.960 − 0.960i)5-s − 1.05i·7-s + (0.706 − 0.707i)9-s + (0.0467 − 0.0467i)11-s + (−0.548 + 0.548i)13-s + (0.519 − 1.25i)15-s − 0.653i·17-s + (−1.11 + 1.11i)19-s + (−0.402 − 0.971i)21-s − 0.339·23-s − 0.846i·25-s + (0.381 − 0.924i)27-s + (1.19 + 1.19i)29-s − 0.802·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.134 + 0.990i$
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: \((2,\ 384,\ (\ :1),\ 0.134 + 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.96555 - 1.71622i\)
\(L(\frac12)\) \(\approx\) \(1.96555 - 1.71622i\)
\(L(2)\) 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 \( 1 + (-2.77 + 1.14i)T \)
good5 \( 1 + (-4.80 + 4.80i)T - 25iT^{2} \)
7 \( 1 + 7.36iT - 49T^{2} \)
11 \( 1 + (-0.514 + 0.514i)T - 121iT^{2} \)
13 \( 1 + (7.12 - 7.12i)T - 169iT^{2} \)
17 \( 1 + 11.1iT - 289T^{2} \)
19 \( 1 + (21.1 - 21.1i)T - 361iT^{2} \)
23 \( 1 + 7.80T + 529T^{2} \)
29 \( 1 + (-34.6 - 34.6i)T + 841iT^{2} \)
31 \( 1 + 24.8T + 961T^{2} \)
37 \( 1 + (-18.2 - 18.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 64.2T + 1.68e3T^{2} \)
43 \( 1 + (-7.24 - 7.24i)T + 1.84e3iT^{2} \)
47 \( 1 + 23.0iT - 2.20e3T^{2} \)
53 \( 1 + (-31.9 + 31.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (17.6 - 17.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-12.3 + 12.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (-41.1 + 41.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 25.6T + 5.04e3T^{2} \)
73 \( 1 + 56.1iT - 5.32e3T^{2} \)
79 \( 1 - 35.7T + 6.24e3T^{2} \)
83 \( 1 + (-94.9 - 94.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 44.8T + 7.92e3T^{2} \)
97 \( 1 + 82.3T + 9.40e3T^{2} \)
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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

−10.65297440177033117242318923341, −9.793294901436978611223054311982, −9.114910547407466906306054835644, −8.213449107035515041692755512551, −7.22864528425831040064821468366, −6.26557145679837503272405183367, −4.85771550635563739343772938633, −3.85332322632133757420167523441, −2.24168396268051400422051806991, −1.10752179809543884291992472412, 2.29439758428989902862446053841, 2.67991513059808555483512634674, 4.26570226121116696337100129804, 5.62290190974627453899776426812, 6.52378779075814712526269341379, 7.71530670947125506360914968396, 8.740478136062748076430243119388, 9.494096967703150041600993127236, 10.30651166629135276559101874252, 11.01444824170816455499958298196

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