Properties

Degree $2$
Conductor $384$
Sign $-0.666 - 0.745i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.23 − 2i)3-s − 4·5-s − 8.94·7-s + (1.00 − 8.94i)9-s − 4.47·11-s + 17.8i·13-s + (−8.94 + 8i)15-s + 17.8i·17-s + 20i·19-s + (−20.0 + 17.8i)21-s − 16i·23-s − 9·25-s + (−15.6 − 22.0i)27-s − 52·29-s + 26.8·31-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)3-s − 0.800·5-s − 1.27·7-s + (0.111 − 0.993i)9-s − 0.406·11-s + 1.37i·13-s + (−0.596 + 0.533i)15-s + 1.05i·17-s + 1.05i·19-s + (−0.952 + 0.851i)21-s − 0.695i·23-s − 0.359·25-s + (−0.579 − 0.814i)27-s − 1.79·29-s + 0.865·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.666 - 0.745i$
Motivic weight: \(2\)
Character: $\chi_{384} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.666 - 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.117923 + 0.263685i\)
\(L(\frac12)\) \(\approx\) \(0.117923 + 0.263685i\)
\(L(2)\) 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 \( 1 + (-2.23 + 2i)T \)
good5 \( 1 + 4T + 25T^{2} \)
7 \( 1 + 8.94T + 49T^{2} \)
11 \( 1 + 4.47T + 121T^{2} \)
13 \( 1 - 17.8iT - 169T^{2} \)
17 \( 1 - 17.8iT - 289T^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 + 16iT - 529T^{2} \)
29 \( 1 + 52T + 841T^{2} \)
31 \( 1 - 26.8T + 961T^{2} \)
37 \( 1 - 53.6iT - 1.36e3T^{2} \)
41 \( 1 + 35.7iT - 1.68e3T^{2} \)
43 \( 1 + 36iT - 1.84e3T^{2} \)
47 \( 1 + 64iT - 2.20e3T^{2} \)
53 \( 1 + 20T + 2.80e3T^{2} \)
59 \( 1 + 102.T + 3.48e3T^{2} \)
61 \( 1 - 17.8iT - 3.72e3T^{2} \)
67 \( 1 - 44iT - 4.48e3T^{2} \)
71 \( 1 + 80iT - 5.04e3T^{2} \)
73 \( 1 + 50T + 5.32e3T^{2} \)
79 \( 1 + 80.4T + 6.24e3T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 - 160. iT - 7.92e3T^{2} \)
97 \( 1 - 50T + 9.40e3T^{2} \)
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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

−11.78712552122569564888758884409, −10.41446256059049957620990387566, −9.497761313841009852342012836776, −8.609854897355980311488507739301, −7.75166623788681679248650177751, −6.81399737835027693146003371190, −6.04363436313573710962835260513, −4.09646598894109535817606068237, −3.38298397368546402677643625056, −1.91729066472765812766389289216, 0.10934780135488740737296090459, 2.80946948900710314451898410473, 3.42638777557875419206071808307, 4.66955066333141990014183063718, 5.84390025854139771137997729019, 7.38292563030178520267310817739, 7.87328150769874482264814309840, 9.198246025281828228910103471490, 9.668148172745897278274573191972, 10.71429015854809730311205602410

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