Properties

Degree $2$
Conductor $384$
Sign $-0.998 - 0.0537i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s + (−6.49 + 6.49i)5-s + 3.94·7-s − 2.99i·9-s + (−4.31 − 4.31i)11-s + (−4.06 − 4.06i)13-s + 15.9i·15-s − 14.5·17-s + (−4.94 + 4.94i)19-s + (4.82 − 4.82i)21-s − 43.6·23-s − 59.3i·25-s + (−3.67 − 3.67i)27-s + (−25.0 − 25.0i)29-s + 32.5i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−1.29 + 1.29i)5-s + 0.563·7-s − 0.333i·9-s + (−0.391 − 0.391i)11-s + (−0.312 − 0.312i)13-s + 1.06i·15-s − 0.856·17-s + (−0.260 + 0.260i)19-s + (0.229 − 0.229i)21-s − 1.89·23-s − 2.37i·25-s + (−0.136 − 0.136i)27-s + (−0.865 − 0.865i)29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0537i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.998 - 0.0537i$
Motivic weight: \(2\)
Character: $\chi_{384} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.998 - 0.0537i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00397473 + 0.147880i\)
\(L(\frac12)\) \(\approx\) \(0.00397473 + 0.147880i\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
good5 \( 1 + (6.49 - 6.49i)T - 25iT^{2} \)
7 \( 1 - 3.94T + 49T^{2} \)
11 \( 1 + (4.31 + 4.31i)T + 121iT^{2} \)
13 \( 1 + (4.06 + 4.06i)T + 169iT^{2} \)
17 \( 1 + 14.5T + 289T^{2} \)
19 \( 1 + (4.94 - 4.94i)T - 361iT^{2} \)
23 \( 1 + 43.6T + 529T^{2} \)
29 \( 1 + (25.0 + 25.0i)T + 841iT^{2} \)
31 \( 1 - 32.5iT - 961T^{2} \)
37 \( 1 + (4.14 - 4.14i)T - 1.36e3iT^{2} \)
41 \( 1 - 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (-16.1 - 16.1i)T + 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (-31.5 + 31.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (-49.7 - 49.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (44.4 + 44.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (-1.64 + 1.64i)T - 4.48e3iT^{2} \)
71 \( 1 - 24.1T + 5.04e3T^{2} \)
73 \( 1 - 10.7iT - 5.32e3T^{2} \)
79 \( 1 + 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (42.0 - 42.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−11.51406012795628391948965901812, −10.84051234705633556197834878590, −9.911649199571104958474860285507, −8.337292929799953744649947974124, −7.932741529637883282993553707396, −7.05936036497507082391573098641, −6.07636667002694595679630100184, −4.39668588746149249455038732950, −3.39624865389310206869797705752, −2.26028907502546226770450896925, 0.05852336110623221592303230759, 2.01314518733153010916561378257, 3.92134247094430773180983008746, 4.46520156709976244175758707209, 5.46124461647598278047025728947, 7.26393093790277236562204155120, 8.032247965688642092050839639532, 8.721087685193230914548084400561, 9.542611947350175557722378193815, 10.79422947104406522684485262806

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