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Properties

Label 2-384-32.21-c1-0-7
Degree $2$
Conductor $384$
Sign $-0.147 + 0.988i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)3-s + (0.705 − 1.70i)5-s + (−3.24 − 3.24i)7-s + (0.707 − 0.707i)9-s + (−3.38 − 1.40i)11-s + (−0.503 − 1.21i)13-s − 1.84i·15-s − 0.622i·17-s + (2.14 + 5.17i)19-s + (−4.23 − 1.75i)21-s + (2.47 − 2.47i)23-s + (1.13 + 1.13i)25-s + (0.382 − 0.923i)27-s + (−2.16 + 0.897i)29-s + 10.4·31-s + ⋯
L(s)  = 1  + (0.533 − 0.220i)3-s + (0.315 − 0.762i)5-s + (−1.22 − 1.22i)7-s + (0.235 − 0.235i)9-s + (−1.02 − 0.423i)11-s + (−0.139 − 0.337i)13-s − 0.476i·15-s − 0.151i·17-s + (0.491 + 1.18i)19-s + (−0.924 − 0.382i)21-s + (0.516 − 0.516i)23-s + (0.226 + 0.226i)25-s + (0.0736 − 0.177i)27-s + (−0.402 + 0.166i)29-s + 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.147 + 0.988i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.147 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.871315 - 1.01138i\)
\(L(\frac12)\) \(\approx\) \(0.871315 - 1.01138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 + 0.382i)T \)
good5 \( 1 + (-0.705 + 1.70i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.24 + 3.24i)T + 7iT^{2} \)
11 \( 1 + (3.38 + 1.40i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.503 + 1.21i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.622iT - 17T^{2} \)
19 \( 1 + (-2.14 - 5.17i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.47 + 2.47i)T - 23iT^{2} \)
29 \( 1 + (2.16 - 0.897i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (0.0714 - 0.172i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-8.50 + 8.50i)T - 41iT^{2} \)
43 \( 1 + (-3.62 - 1.50i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 5.02iT - 47T^{2} \)
53 \( 1 + (7.15 + 2.96i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.52 + 3.68i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.07 - 1.27i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-2.17 + 0.901i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.11 - 1.11i)T + 71iT^{2} \)
73 \( 1 + (-3.71 + 3.71i)T - 73iT^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + (-4.69 - 11.3i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.54 + 3.54i)T + 89iT^{2} \)
97 \( 1 - 0.139T + 97T^{2} \)
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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

−10.82423117934566306212380852075, −10.07342845999047413134696410902, −9.364685016844974211663849403086, −8.239930074767535175782349084199, −7.44886516548956788042446833261, −6.38645014400651757707569485930, −5.22320042651549348122842539848, −3.88217841139238467046962863358, −2.81428544617093241523203010771, −0.842993798610073370112353680567, 2.59720542308778751777539657441, 2.94483986303885731203335079788, 4.71728775616819799348881466963, 5.94385757870580015846784221202, 6.80493951420777498398618085261, 7.88088633498203363064705230440, 9.124270392046542124053217431035, 9.622062345481851470587914285592, 10.49425283539525992712929484642, 11.55114687142120845930056560332

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