Properties

Label 2-384-128.101-c1-0-14
Degree $2$
Conductor $384$
Sign $0.189 - 0.981i$
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  + (0.208 + 1.39i)2-s + (0.881 − 0.471i)3-s + (−1.91 + 0.583i)4-s + (1.48 + 1.81i)5-s + (0.843 + 1.13i)6-s + (2.68 − 1.79i)7-s + (−1.21 − 2.55i)8-s + (0.555 − 0.831i)9-s + (−2.22 + 2.46i)10-s + (−0.671 + 0.203i)11-s + (−1.41 + 1.41i)12-s + (4.24 + 3.48i)13-s + (3.06 + 3.37i)14-s + (2.16 + 0.897i)15-s + (3.31 − 2.23i)16-s + (−4.31 + 1.78i)17-s + ⋯
L(s)  = 1  + (0.147 + 0.989i)2-s + (0.509 − 0.272i)3-s + (−0.956 + 0.291i)4-s + (0.665 + 0.810i)5-s + (0.344 + 0.463i)6-s + (1.01 − 0.677i)7-s + (−0.429 − 0.902i)8-s + (0.185 − 0.277i)9-s + (−0.703 + 0.777i)10-s + (−0.202 + 0.0613i)11-s + (−0.407 + 0.408i)12-s + (1.17 + 0.966i)13-s + (0.819 + 0.902i)14-s + (0.559 + 0.231i)15-s + (0.829 − 0.558i)16-s + (−1.04 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43767 + 1.18691i\)
\(L(\frac12)\) \(\approx\) \(1.43767 + 1.18691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.208 - 1.39i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (-1.48 - 1.81i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-2.68 + 1.79i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.671 - 0.203i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-4.24 - 3.48i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (4.31 - 1.78i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.0391 + 0.397i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (2.48 - 0.493i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (0.449 - 1.48i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (3.23 + 3.23i)T + 31iT^{2} \)
37 \( 1 + (6.20 + 0.610i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-1.22 - 6.16i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-10.2 - 5.46i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (3.46 + 8.36i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (2.45 + 8.09i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-2.49 + 2.04i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (5.58 + 10.4i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (6.84 + 12.8i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-7.22 - 10.8i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (4.10 + 2.74i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-3.49 + 8.42i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-2.18 + 0.215i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (0.00810 + 0.00161i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-2.67 - 2.67i)T + 97iT^{2} \)
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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

−11.38424495416166485934362490744, −10.59775278101764319642885406258, −9.456569355153846198333776309828, −8.558514305712053142842933187298, −7.73711492255292472208412177207, −6.75912331925927530997555317835, −6.12671623983782729093842523636, −4.66474861560179859820704936458, −3.66610014389958856177648580508, −1.86592134860658508485805416463, 1.46348744475586084042342821716, 2.59424953665795233192429836215, 4.04527997053083765582383986303, 5.13445313434952978737350299041, 5.79128718695626271893366889980, 7.88549259299796070871337832306, 8.905029663690181280872102223498, 9.009381781812360122700672740994, 10.41868349752258103226732388701, 11.00023672527215181418323612398

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