Properties

Degree $2$
Conductor $384$
Sign $0.439 - 0.898i$
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 + (−3.32 − 3.32i)5-s − 4.04·7-s + 2.99i·9-s + (−6.82 + 6.82i)11-s + (−4.29 + 4.29i)13-s + 8.14i·15-s + 30.1·17-s + (19.7 + 19.7i)19-s + (4.94 + 4.94i)21-s − 28.2·23-s − 2.86i·25-s + (3.67 − 3.67i)27-s + (21.3 − 21.3i)29-s + 38.0i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.665 − 0.665i)5-s − 0.577·7-s + 0.333i·9-s + (−0.620 + 0.620i)11-s + (−0.330 + 0.330i)13-s + 0.543i·15-s + 1.77·17-s + (1.03 + 1.03i)19-s + (0.235 + 0.235i)21-s − 1.22·23-s − 0.114i·25-s + (0.136 − 0.136i)27-s + (0.736 − 0.736i)29-s + 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.439 - 0.898i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.439 - 0.898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.651900 + 0.406612i\)
\(L(\frac12)\) \(\approx\) \(0.651900 + 0.406612i\)
\(L(2)\) 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 \( 1 + (3.32 + 3.32i)T + 25iT^{2} \)
7 \( 1 + 4.04T + 49T^{2} \)
11 \( 1 + (6.82 - 6.82i)T - 121iT^{2} \)
13 \( 1 + (4.29 - 4.29i)T - 169iT^{2} \)
17 \( 1 - 30.1T + 289T^{2} \)
19 \( 1 + (-19.7 - 19.7i)T + 361iT^{2} \)
23 \( 1 + 28.2T + 529T^{2} \)
29 \( 1 + (-21.3 + 21.3i)T - 841iT^{2} \)
31 \( 1 - 38.0iT - 961T^{2} \)
37 \( 1 + (-42.8 - 42.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (32.6 - 32.6i)T - 1.84e3iT^{2} \)
47 \( 1 + 15.8iT - 2.20e3T^{2} \)
53 \( 1 + (-0.476 - 0.476i)T + 2.80e3iT^{2} \)
59 \( 1 + (9.97 - 9.97i)T - 3.48e3iT^{2} \)
61 \( 1 + (37.9 - 37.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (20.0 + 20.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 40.0T + 5.04e3T^{2} \)
73 \( 1 - 30.8iT - 5.32e3T^{2} \)
79 \( 1 - 130. iT - 6.24e3T^{2} \)
83 \( 1 + (-2.26 - 2.26i)T + 6.88e3iT^{2} \)
89 \( 1 + 72.2iT - 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−11.68903998119999375257340065226, −10.06893347757862927679808072237, −9.814270716992870063264542868857, −8.082470460683749508106941666490, −7.81779131249690839546085526938, −6.51562315415582614488824577923, −5.46117163732130582058679500570, −4.44615589269296858794040512335, −3.08019644782541679824850538373, −1.24591154495586678913752955909, 0.40412861978402106837040089947, 2.92461643739735859205707570619, 3.69893352614615021226328209399, 5.19421550049510685529757677071, 6.03972856051002819096608328519, 7.30966825392934412988434897605, 7.948833514814111541122058251656, 9.370347325130695753685012867212, 10.13965811375034492559082927057, 10.93910426238871212771491111008

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