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Properties

Label 2-384-32.21-c1-0-0
Degree $2$
Conductor $384$
Sign $0.526 - 0.850i$
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 + (−0.155 + 0.375i)5-s + (−0.709 − 0.709i)7-s + (0.707 − 0.707i)9-s + (2.79 + 1.15i)11-s + (2.58 + 6.24i)13-s − 0.406i·15-s − 1.05i·17-s + (1.48 + 3.59i)19-s + (0.926 + 0.383i)21-s + (−0.922 + 0.922i)23-s + (3.41 + 3.41i)25-s + (−0.382 + 0.923i)27-s + (−7.64 + 3.16i)29-s + 1.88·31-s + ⋯
L(s)  = 1  + (−0.533 + 0.220i)3-s + (−0.0696 + 0.168i)5-s + (−0.268 − 0.268i)7-s + (0.235 − 0.235i)9-s + (0.841 + 0.348i)11-s + (0.717 + 1.73i)13-s − 0.105i·15-s − 0.256i·17-s + (0.341 + 0.823i)19-s + (0.202 + 0.0837i)21-s + (−0.192 + 0.192i)23-s + (0.683 + 0.683i)25-s + (−0.0736 + 0.177i)27-s + (−1.41 + 0.587i)29-s + 0.338·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.526 - 0.850i$
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: \((2,\ 384,\ (\ :1/2),\ 0.526 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975448 + 0.543621i\)
\(L(\frac12)\) \(\approx\) \(0.975448 + 0.543621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.155 - 0.375i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.709 + 0.709i)T + 7iT^{2} \)
11 \( 1 + (-2.79 - 1.15i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-2.58 - 6.24i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 1.05iT - 17T^{2} \)
19 \( 1 + (-1.48 - 3.59i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.922 - 0.922i)T - 23iT^{2} \)
29 \( 1 + (7.64 - 3.16i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.88T + 31T^{2} \)
37 \( 1 + (-1.24 + 3.01i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.11 + 5.11i)T - 41iT^{2} \)
43 \( 1 + (-10.9 - 4.53i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.47iT - 47T^{2} \)
53 \( 1 + (7.58 + 3.14i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.13 + 9.98i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.35 + 0.562i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (10.8 - 4.50i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (9.35 + 9.35i)T + 71iT^{2} \)
73 \( 1 + (-0.367 + 0.367i)T - 73iT^{2} \)
79 \( 1 + 5.87iT - 79T^{2} \)
83 \( 1 + (1.62 + 3.91i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.33 - 8.33i)T + 89iT^{2} \)
97 \( 1 + 7.97T + 97T^{2} \)
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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

−11.40349628695803343272874106230, −10.77927351644147064530585551820, −9.481492358553596414708308530045, −9.111884875953572638629682594812, −7.52464195622212835077589590872, −6.70164683558975792023926618660, −5.82964934035644000304675444583, −4.42988475535579779861080366408, −3.61187794317329640844565159140, −1.59144448211380388815415720723, 0.898108215976303428298855185036, 2.88449038527195671075572442095, 4.22282252117459278078737449326, 5.60228531050398567255081533684, 6.20643907215237427971850768197, 7.42467888069418979619796120585, 8.415660478141229334906903002056, 9.334105495251003276094170280219, 10.46431108341553171868350793813, 11.17603624031264785729270651351

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