Properties

Label 2-384-384.275-c1-0-11
Degree $2$
Conductor $384$
Sign $-0.135 - 0.990i$
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  + (−1.22 − 0.711i)2-s + (0.885 + 1.48i)3-s + (0.987 + 1.73i)4-s + (1.29 + 1.06i)5-s + (−0.0239 − 2.44i)6-s + (−1.87 + 2.80i)7-s + (0.0294 − 2.82i)8-s + (−1.43 + 2.63i)9-s + (−0.829 − 2.22i)10-s + (1.89 + 0.575i)11-s + (−1.71 + 3.01i)12-s + (−1.57 + 1.29i)13-s + (4.29 − 2.09i)14-s + (−0.436 + 2.87i)15-s + (−2.04 + 3.43i)16-s + (1.97 − 4.76i)17-s + ⋯
L(s)  = 1  + (−0.864 − 0.503i)2-s + (0.511 + 0.859i)3-s + (0.493 + 0.869i)4-s + (0.581 + 0.476i)5-s + (−0.00979 − 0.999i)6-s + (−0.709 + 1.06i)7-s + (0.0104 − 0.999i)8-s + (−0.476 + 0.878i)9-s + (−0.262 − 0.704i)10-s + (0.572 + 0.173i)11-s + (−0.494 + 0.869i)12-s + (−0.437 + 0.359i)13-s + (1.14 − 0.560i)14-s + (−0.112 + 0.743i)15-s + (−0.511 + 0.859i)16-s + (0.478 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.135 - 0.990i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.135 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653367 + 0.749039i\)
\(L(\frac12)\) \(\approx\) \(0.653367 + 0.749039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.711i)T \)
3 \( 1 + (-0.885 - 1.48i)T \)
good5 \( 1 + (-1.29 - 1.06i)T + (0.975 + 4.90i)T^{2} \)
7 \( 1 + (1.87 - 2.80i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.89 - 0.575i)T + (9.14 + 6.11i)T^{2} \)
13 \( 1 + (1.57 - 1.29i)T + (2.53 - 12.7i)T^{2} \)
17 \( 1 + (-1.97 + 4.76i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (6.12 + 0.603i)T + (18.6 + 3.70i)T^{2} \)
23 \( 1 + (-0.931 - 0.185i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (6.17 - 1.87i)T + (24.1 - 16.1i)T^{2} \)
31 \( 1 + (-6.66 - 6.66i)T + 31iT^{2} \)
37 \( 1 + (-10.3 + 1.01i)T + (36.2 - 7.21i)T^{2} \)
41 \( 1 + (-0.0648 - 0.0129i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-3.83 - 7.16i)T + (-23.8 + 35.7i)T^{2} \)
47 \( 1 + (0.424 - 1.02i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (-11.1 - 3.37i)T + (44.0 + 29.4i)T^{2} \)
59 \( 1 + (3.82 + 3.14i)T + (11.5 + 57.8i)T^{2} \)
61 \( 1 + (-1.52 + 2.85i)T + (-33.8 - 50.7i)T^{2} \)
67 \( 1 + (-5.71 - 3.05i)T + (37.2 + 55.7i)T^{2} \)
71 \( 1 + (2.87 - 4.29i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (5.34 - 3.57i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-1.90 + 0.788i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-13.7 - 1.35i)T + (81.4 + 16.1i)T^{2} \)
89 \( 1 + (2.14 + 10.7i)T + (-82.2 + 34.0i)T^{2} \)
97 \( 1 + (-5.47 + 5.47i)T - 97iT^{2} \)
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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

−11.36307649591939733825614871987, −10.41552400831863834087276661566, −9.552842861844110324524704760954, −9.249982053428285595637257898163, −8.284030260996295717047084284688, −6.98745489916770944889568689528, −5.97117085067191938196699882982, −4.40686059691586756335949230810, −2.98061310139697031195396233799, −2.31839338366476895015102339975, 0.818832287580987839865872434658, 2.18152846512932433525696043602, 3.94207491575390658651483074160, 5.84191604665086233314583839491, 6.45510827239770522875062730447, 7.45742740857835748885544502858, 8.225727323154297579287315686439, 9.199217549471825054211662372088, 9.902660656968917551717716703763, 10.80228839893922026942085365002

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