Properties

Label 2-384-3.2-c2-0-4
Degree $2$
Conductor $384$
Sign $0.442 - 0.896i$
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  + (−2.69 − 1.32i)3-s + 0.640i·5-s − 2.72·7-s + (5.47 + 7.14i)9-s − 11.2i·11-s − 5.25·13-s + (0.849 − 1.72i)15-s + 14.8i·17-s − 15.0·19-s + (7.31 + 3.61i)21-s + 36.4i·23-s + 24.5·25-s + (−5.24 − 26.4i)27-s + 51.7i·29-s + 36.5·31-s + ⋯
L(s)  = 1  + (−0.896 − 0.442i)3-s + 0.128i·5-s − 0.388·7-s + (0.608 + 0.793i)9-s − 1.02i·11-s − 0.403·13-s + (0.0566 − 0.114i)15-s + 0.874i·17-s − 0.793·19-s + (0.348 + 0.171i)21-s + 1.58i·23-s + 0.983·25-s + (−0.194 − 0.980i)27-s + 1.78i·29-s + 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.442 - 0.896i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.442 - 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.719861 + 0.447521i\)
\(L(\frac12)\) \(\approx\) \(0.719861 + 0.447521i\)
\(L(2)\) 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 \( 1 + (2.69 + 1.32i)T \)
good5 \( 1 - 0.640iT - 25T^{2} \)
7 \( 1 + 2.72T + 49T^{2} \)
11 \( 1 + 11.2iT - 121T^{2} \)
13 \( 1 + 5.25T + 169T^{2} \)
17 \( 1 - 14.8iT - 289T^{2} \)
19 \( 1 + 15.0T + 361T^{2} \)
23 \( 1 - 36.4iT - 529T^{2} \)
29 \( 1 - 51.7iT - 841T^{2} \)
31 \( 1 - 36.5T + 961T^{2} \)
37 \( 1 - 63.6T + 1.36e3T^{2} \)
41 \( 1 + 12.1iT - 1.68e3T^{2} \)
43 \( 1 - 11.8T + 1.84e3T^{2} \)
47 \( 1 - 61.1iT - 2.20e3T^{2} \)
53 \( 1 - 59.1iT - 2.80e3T^{2} \)
59 \( 1 - 37.2iT - 3.48e3T^{2} \)
61 \( 1 + 58.1T + 3.72e3T^{2} \)
67 \( 1 + 23.0T + 4.48e3T^{2} \)
71 \( 1 - 7.29iT - 5.04e3T^{2} \)
73 \( 1 - 73.4T + 5.32e3T^{2} \)
79 \( 1 + 58.5T + 6.24e3T^{2} \)
83 \( 1 - 32.3iT - 6.88e3T^{2} \)
89 \( 1 + 112. iT - 7.92e3T^{2} \)
97 \( 1 + 80.0T + 9.40e3T^{2} \)
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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

−11.12290186145492377725693551465, −10.70134410814730963340852147383, −9.565743641024105546155531986115, −8.414068154331569396266349114698, −7.40014638399134641602099388123, −6.38773535243231846780466130537, −5.70947820939980192370748541091, −4.47956539728406419654575353720, −3.00459792155750266339613860987, −1.24387185107191800511290707845, 0.47177555516465047279712054696, 2.53895069861475347375307809329, 4.30068384479366205265991068582, 4.87424113827687934950933352967, 6.25777923558024939517116824114, 6.88831921757974554297406893443, 8.179361072908701418925248128291, 9.506745631900417763508214486168, 9.982872560451013071550814391347, 10.93586047090646517262649691389

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