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Properties

Label 2-384-48.5-c2-0-3
Degree $2$
Conductor $384$
Sign $-0.843 + 0.537i$
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  + (1.13 + 2.77i)3-s + (−6.28 + 6.28i)5-s − 1.64i·7-s + (−6.43 + 6.29i)9-s + (−4.75 + 4.75i)11-s + (9.35 − 9.35i)13-s + (−24.5 − 10.3i)15-s − 11.4i·17-s + (−8.58 + 8.58i)19-s + (4.57 − 1.86i)21-s − 16.2·23-s − 54.0i·25-s + (−24.7 − 10.7i)27-s + (10.7 + 10.7i)29-s − 6.35·31-s + ⋯
L(s)  = 1  + (0.377 + 0.926i)3-s + (−1.25 + 1.25i)5-s − 0.235i·7-s + (−0.715 + 0.699i)9-s + (−0.432 + 0.432i)11-s + (0.719 − 0.719i)13-s + (−1.63 − 0.689i)15-s − 0.675i·17-s + (−0.451 + 0.451i)19-s + (0.217 − 0.0887i)21-s − 0.706·23-s − 2.16i·25-s + (−0.917 − 0.398i)27-s + (0.370 + 0.370i)29-s − 0.204·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.843 + 0.537i$
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: \((2,\ 384,\ (\ :1),\ -0.843 + 0.537i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.160058 - 0.548634i\)
\(L(\frac12)\) \(\approx\) \(0.160058 - 0.548634i\)
\(L(2)\) 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 \( 1 + (-1.13 - 2.77i)T \)
good5 \( 1 + (6.28 - 6.28i)T - 25iT^{2} \)
7 \( 1 + 1.64iT - 49T^{2} \)
11 \( 1 + (4.75 - 4.75i)T - 121iT^{2} \)
13 \( 1 + (-9.35 + 9.35i)T - 169iT^{2} \)
17 \( 1 + 11.4iT - 289T^{2} \)
19 \( 1 + (8.58 - 8.58i)T - 361iT^{2} \)
23 \( 1 + 16.2T + 529T^{2} \)
29 \( 1 + (-10.7 - 10.7i)T + 841iT^{2} \)
31 \( 1 + 6.35T + 961T^{2} \)
37 \( 1 + (27.2 + 27.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 1.98T + 1.68e3T^{2} \)
43 \( 1 + (19.4 + 19.4i)T + 1.84e3iT^{2} \)
47 \( 1 - 74.9iT - 2.20e3T^{2} \)
53 \( 1 + (4.00 - 4.00i)T - 2.80e3iT^{2} \)
59 \( 1 + (27.9 - 27.9i)T - 3.48e3iT^{2} \)
61 \( 1 + (39.2 - 39.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (68.6 - 68.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 - 59.0iT - 5.32e3T^{2} \)
79 \( 1 + 17.3T + 6.24e3T^{2} \)
83 \( 1 + (75.1 + 75.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 78.8T + 7.92e3T^{2} \)
97 \( 1 + 38.8T + 9.40e3T^{2} \)
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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

−11.39179961407343040361994791510, −10.58386457488043441479230817237, −10.22888259804057721098837739519, −8.831222434663301906140565683537, −7.893048806281871579382313591290, −7.23801698742031227272891799176, −5.87561369320960622742279460108, −4.45180634594873499385390140718, −3.59826512079401137226247024826, −2.71085451786331077872787149087, 0.23824842545364338831095578937, 1.67178154756992308866850399460, 3.41968031416499195340733035861, 4.47613454595352772615037930208, 5.79834008517344163043791374512, 6.92355325476161494316406867520, 8.133932127021257415977537819283, 8.404209772332629167285806565096, 9.241536165886290282259788401347, 10.86467946252222481311540738683

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