Properties

Label 2-384-8.3-c6-0-33
Degree $2$
Conductor $384$
Sign $i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.5·3-s + 20i·5-s + 155. i·7-s + 243·9-s − 2.30e3·11-s + 3.22e3i·13-s + 311. i·15-s − 6.56e3·17-s + 3.92e3·19-s + 2.42e3i·21-s − 1.73e4i·23-s + 1.52e4·25-s + 3.78e3·27-s − 4.47e4i·29-s + 3.06e4i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.160i·5-s + 0.454i·7-s + 0.333·9-s − 1.73·11-s + 1.46i·13-s + 0.0923i·15-s − 1.33·17-s + 0.572·19-s + 0.262i·21-s − 1.42i·23-s + 0.974·25-s + 0.192·27-s − 1.83i·29-s + 1.02i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.172084107\)
\(L(\frac12)\) \(\approx\) \(1.172084107\)
\(L(4)\) 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 \( 1 - 15.5T \)
good5 \( 1 - 20iT - 1.56e4T^{2} \)
7 \( 1 - 155. iT - 1.17e5T^{2} \)
11 \( 1 + 2.30e3T + 1.77e6T^{2} \)
13 \( 1 - 3.22e3iT - 4.82e6T^{2} \)
17 \( 1 + 6.56e3T + 2.41e7T^{2} \)
19 \( 1 - 3.92e3T + 4.70e7T^{2} \)
23 \( 1 + 1.73e4iT - 1.48e8T^{2} \)
29 \( 1 + 4.47e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.06e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.41e4iT - 2.56e9T^{2} \)
41 \( 1 - 9.00e3T + 4.75e9T^{2} \)
43 \( 1 + 2.52e4T + 6.32e9T^{2} \)
47 \( 1 + 1.75e5iT - 1.07e10T^{2} \)
53 \( 1 - 9.62e4iT - 2.21e10T^{2} \)
59 \( 1 - 3.50e4T + 4.21e10T^{2} \)
61 \( 1 + 8.61e4iT - 5.15e10T^{2} \)
67 \( 1 - 4.24e5T + 9.04e10T^{2} \)
71 \( 1 + 3.65e4iT - 1.28e11T^{2} \)
73 \( 1 + 3.75e5T + 1.51e11T^{2} \)
79 \( 1 + 5.20e5iT - 2.43e11T^{2} \)
83 \( 1 + 5.94e5T + 3.26e11T^{2} \)
89 \( 1 - 9.01e5T + 4.96e11T^{2} \)
97 \( 1 + 1.25e6T + 8.32e11T^{2} \)
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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

−10.11777183586637945108902788739, −9.003462346386623654841449803942, −8.428933682126227842385380298405, −7.30601346985704866631905527000, −6.44441896204282826724971970436, −5.10863541598315236194614165245, −4.19330390412043464449932442340, −2.68315984098208017332870987040, −2.12003606621360491855343777696, −0.25292739133215055809237351601, 1.04988784972774636629409630315, 2.55709092696575739361760104457, 3.35803496105749866866115515188, 4.79511067321974240198204061123, 5.57494729999077230555562229734, 7.09270205406052130243783669944, 7.81220367811376445095626079016, 8.591316573801081200065568195854, 9.701721159413026310069349610363, 10.54312038813479200329867768311

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