Properties

Degree $2$
Conductor $384$
Sign $0.985 + 0.169i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + i)3-s + 2.82·5-s − 2.82i·7-s + (1.00 − 2.82i)9-s − 2i·11-s + 4i·13-s + (−4.00 + 2.82i)15-s − 5.65i·17-s + 2.82·19-s + (2.82 + 4.00i)21-s + 8·23-s + 3.00·25-s + (1.41 + 5.00i)27-s + 2.82·29-s + 8.48i·31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + 1.26·5-s − 1.06i·7-s + (0.333 − 0.942i)9-s − 0.603i·11-s + 1.10i·13-s + (−1.03 + 0.730i)15-s − 1.37i·17-s + 0.648·19-s + (0.617 + 0.872i)21-s + 1.66·23-s + 0.600·25-s + (0.272 + 0.962i)27-s + 0.525·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.985 + 0.169i$
Motivic weight: \(1\)
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30749 - 0.111352i\)
\(L(\frac12)\) \(\approx\) \(1.30749 - 0.111352i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
good5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 5.65iT - 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.48T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 2.82iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−11.12000131174485754522757097781, −10.46384748469546298881043016343, −9.547300273663319392966527971149, −9.014640719020964914143683714319, −7.15802566357469675757158148371, −6.56216569056071767764736280298, −5.38132132023293670107484485591, −4.62376623683424415542603396228, −3.17345858220063849604074900420, −1.15882525896736858807680651954, 1.54725662438713011073558818112, 2.73251475822297227269246547105, 4.90262128472470923969857010055, 5.73925680093140389530464301458, 6.27151545042709666373914242059, 7.51722692481107305037871800976, 8.613239515071452055373031359784, 9.714848876048339711521911435546, 10.42540176342404056017053440804, 11.39710785491590678053214760678

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