Properties

Label 2-384-48.5-c2-0-10
Degree $2$
Conductor $384$
Sign $-0.509 - 0.860i$
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  + (−0.164 + 2.99i)3-s + (3.61 − 3.61i)5-s + 12.2i·7-s + (−8.94 − 0.985i)9-s + (1.76 − 1.76i)11-s + (2.38 − 2.38i)13-s + (10.2 + 11.4i)15-s + 20.0i·17-s + (−8.77 + 8.77i)19-s + (−36.7 − 2.02i)21-s + 13.1·23-s − 1.10i·25-s + (4.42 − 26.6i)27-s + (6.51 + 6.51i)29-s − 37.5·31-s + ⋯
L(s)  = 1  + (−0.0548 + 0.998i)3-s + (0.722 − 0.722i)5-s + 1.75i·7-s + (−0.993 − 0.109i)9-s + (0.160 − 0.160i)11-s + (0.183 − 0.183i)13-s + (0.681 + 0.761i)15-s + 1.18i·17-s + (−0.461 + 0.461i)19-s + (−1.75 − 0.0962i)21-s + 0.573·23-s − 0.0443i·25-s + (0.163 − 0.986i)27-s + (0.224 + 0.224i)29-s − 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.778562 + 1.36583i\)
\(L(\frac12)\) \(\approx\) \(0.778562 + 1.36583i\)
\(L(2)\) 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 \( 1 + (0.164 - 2.99i)T \)
good5 \( 1 + (-3.61 + 3.61i)T - 25iT^{2} \)
7 \( 1 - 12.2iT - 49T^{2} \)
11 \( 1 + (-1.76 + 1.76i)T - 121iT^{2} \)
13 \( 1 + (-2.38 + 2.38i)T - 169iT^{2} \)
17 \( 1 - 20.0iT - 289T^{2} \)
19 \( 1 + (8.77 - 8.77i)T - 361iT^{2} \)
23 \( 1 - 13.1T + 529T^{2} \)
29 \( 1 + (-6.51 - 6.51i)T + 841iT^{2} \)
31 \( 1 + 37.5T + 961T^{2} \)
37 \( 1 + (10.0 + 10.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 4.57T + 1.68e3T^{2} \)
43 \( 1 + (-21.2 - 21.2i)T + 1.84e3iT^{2} \)
47 \( 1 - 54.8iT - 2.20e3T^{2} \)
53 \( 1 + (21.5 - 21.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (53.6 - 53.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-19.2 + 19.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-31.5 + 31.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 65.1T + 5.04e3T^{2} \)
73 \( 1 + 50.2iT - 5.32e3T^{2} \)
79 \( 1 + 20.9T + 6.24e3T^{2} \)
83 \( 1 + (6.35 + 6.35i)T + 6.88e3iT^{2} \)
89 \( 1 - 166.T + 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−11.31909904593942509809781738491, −10.47348029495128280700386399934, −9.295945494660093012285155564836, −9.006497243133383432100829133654, −8.159032360496697097417598435801, −6.09534550117944762102092643151, −5.67506165169450842323283740121, −4.71370304861812923287699910047, −3.28723435812381230977056815541, −1.90884695544797960554063108634, 0.69448181364150815502386157117, 2.15472249966132453222257658749, 3.50651765729263073551908273348, 4.99477223623201310937933706246, 6.42173812958018216631769504780, 6.98113618603336053087853205340, 7.64154064539392185038024164167, 8.979686743523189835471117083019, 10.08049598009515760905509196988, 10.85977436284967948762987928290

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