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Properties

Label 2-384-48.5-c2-0-14
Degree $2$
Conductor $384$
Sign $0.990 - 0.134i$
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  + (−2.77 + 1.14i)3-s + (4.80 − 4.80i)5-s + 7.36i·7-s + (6.35 − 6.37i)9-s + (−0.514 + 0.514i)11-s + (−7.12 + 7.12i)13-s + (−7.79 + 18.8i)15-s − 11.1i·17-s + (21.1 − 21.1i)19-s + (−8.46 − 20.4i)21-s + 7.80·23-s − 21.1i·25-s + (−10.2 + 24.9i)27-s + (34.6 + 34.6i)29-s + 24.8·31-s + ⋯
L(s)  = 1  + (−0.923 + 0.383i)3-s + (0.960 − 0.960i)5-s + 1.05i·7-s + (0.706 − 0.707i)9-s + (−0.0467 + 0.0467i)11-s + (−0.548 + 0.548i)13-s + (−0.519 + 1.25i)15-s − 0.653i·17-s + (1.11 − 1.11i)19-s + (−0.402 − 0.971i)21-s + 0.339·23-s − 0.846i·25-s + (−0.381 + 0.924i)27-s + (1.19 + 1.19i)29-s + 0.802·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.990 - 0.134i$
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: \((2,\ 384,\ (\ :1),\ 0.990 - 0.134i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49642 + 0.101336i\)
\(L(\frac12)\) \(\approx\) \(1.49642 + 0.101336i\)
\(L(2)\) 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 \( 1 + (2.77 - 1.14i)T \)
good5 \( 1 + (-4.80 + 4.80i)T - 25iT^{2} \)
7 \( 1 - 7.36iT - 49T^{2} \)
11 \( 1 + (0.514 - 0.514i)T - 121iT^{2} \)
13 \( 1 + (7.12 - 7.12i)T - 169iT^{2} \)
17 \( 1 + 11.1iT - 289T^{2} \)
19 \( 1 + (-21.1 + 21.1i)T - 361iT^{2} \)
23 \( 1 - 7.80T + 529T^{2} \)
29 \( 1 + (-34.6 - 34.6i)T + 841iT^{2} \)
31 \( 1 - 24.8T + 961T^{2} \)
37 \( 1 + (-18.2 - 18.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 64.2T + 1.68e3T^{2} \)
43 \( 1 + (7.24 + 7.24i)T + 1.84e3iT^{2} \)
47 \( 1 - 23.0iT - 2.20e3T^{2} \)
53 \( 1 + (-31.9 + 31.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (-17.6 + 17.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-12.3 + 12.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (41.1 - 41.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 25.6T + 5.04e3T^{2} \)
73 \( 1 + 56.1iT - 5.32e3T^{2} \)
79 \( 1 + 35.7T + 6.24e3T^{2} \)
83 \( 1 + (94.9 + 94.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 44.8T + 7.92e3T^{2} \)
97 \( 1 + 82.3T + 9.40e3T^{2} \)
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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

−11.32219098470020344336697642305, −10.04304644750327320940292070215, −9.348415897255778197084250543848, −8.809432619359926063872413299837, −7.15111917252825376298953417456, −6.10282816163791495434776134316, −5.16368506042966750851649961941, −4.73560493362841777202310307774, −2.68147674049590542182389098348, −1.05439132120166771931396458118, 1.04000172437002276387750193555, 2.60083414499535883330992889191, 4.18749979888706302992327236347, 5.56464487586994213717872824432, 6.27703104634439387407012874038, 7.20457472120678859492424867892, 7.949899870396699064118446420388, 9.857561424421955318771607695129, 10.20899035020905261383323919378, 10.92846433620192330119263704745

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