Properties

Degree $2$
Conductor $384$
Sign $-0.356 + 0.934i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 0.618i)3-s + 1.23i·5-s − 3.23i·7-s + (2.23 + 2.00i)9-s + 0.763·11-s − 4.47·13-s + (0.763 − 2.00i)15-s − 6.47i·17-s − 5.23i·19-s + (−2.00 + 5.23i)21-s − 6.47·23-s + 3.47·25-s + (−2.38 − 4.61i)27-s − 9.23i·29-s + 0.763i·31-s + ⋯
L(s)  = 1  + (−0.934 − 0.356i)3-s + 0.552i·5-s − 1.22i·7-s + (0.745 + 0.666i)9-s + 0.230·11-s − 1.24·13-s + (0.197 − 0.516i)15-s − 1.56i·17-s − 1.20i·19-s + (−0.436 + 1.14i)21-s − 1.34·23-s + 0.694·25-s + (−0.458 − 0.888i)27-s − 1.71i·29-s + 0.137i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.356 + 0.934i$
Motivic weight: \(1\)
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.356 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405237 - 0.588580i\)
\(L(\frac12)\) \(\approx\) \(0.405237 - 0.588580i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.618i)T \)
good5 \( 1 - 1.23iT - 5T^{2} \)
7 \( 1 + 3.23iT - 7T^{2} \)
11 \( 1 - 0.763T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 6.47iT - 17T^{2} \)
19 \( 1 + 5.23iT - 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + 9.23iT - 29T^{2} \)
31 \( 1 - 0.763iT - 31T^{2} \)
37 \( 1 - 0.472T + 37T^{2} \)
41 \( 1 - 2.47iT - 41T^{2} \)
43 \( 1 - 2.76iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 - 3.23T + 59T^{2} \)
61 \( 1 - 8.47T + 61T^{2} \)
67 \( 1 - 3.70iT - 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 7.23T + 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 + 8.47T + 97T^{2} \)
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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

−11.26211072306156908490365794355, −10.14678199095696782443783019204, −9.660873371651913088279251927592, −7.85984494523370398543395636615, −7.11598315438957270778129682342, −6.55095056031657267483702077158, −5.11666821562140884642663997519, −4.26104731231808745258099638198, −2.52866925710650984195584796654, −0.52833814300405234554427011740, 1.83548621488853150549222289154, 3.74678272242361289903181249756, 4.98790155860994296026882195668, 5.70087326337340059556178010635, 6.61433965144405487198754339378, 8.036084077565844897833599344961, 8.959217521451150062476339426517, 9.896793546833873198011606457929, 10.64238712966067082987870867527, 11.87980248910160989462261459127

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