Properties

Label 2-384-128.101-c1-0-20
Degree $2$
Conductor $384$
Sign $0.958 + 0.285i$
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  + (−0.127 + 1.40i)2-s + (0.881 − 0.471i)3-s + (−1.96 − 0.359i)4-s + (−1.40 − 1.71i)5-s + (0.551 + 1.30i)6-s + (0.0127 − 0.00855i)7-s + (0.758 − 2.72i)8-s + (0.555 − 0.831i)9-s + (2.59 − 1.76i)10-s + (3.40 − 1.03i)11-s + (−1.90 + 0.610i)12-s + (−2.41 − 1.98i)13-s + (0.0104 + 0.0191i)14-s + (−2.04 − 0.848i)15-s + (3.74 + 1.41i)16-s + (4.96 − 2.05i)17-s + ⋯
L(s)  = 1  + (−0.0903 + 0.995i)2-s + (0.509 − 0.272i)3-s + (−0.983 − 0.179i)4-s + (−0.629 − 0.766i)5-s + (0.225 + 0.531i)6-s + (0.00483 − 0.00323i)7-s + (0.268 − 0.963i)8-s + (0.185 − 0.277i)9-s + (0.820 − 0.557i)10-s + (1.02 − 0.311i)11-s + (−0.549 + 0.176i)12-s + (−0.670 − 0.550i)13-s + (0.00278 + 0.00510i)14-s + (−0.529 − 0.219i)15-s + (0.935 + 0.353i)16-s + (1.20 − 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.958 + 0.285i$
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: \((2,\ 384,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21632 - 0.177328i\)
\(L(\frac12)\) \(\approx\) \(1.21632 - 0.177328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.127 - 1.40i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (1.40 + 1.71i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-0.0127 + 0.00855i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-3.40 + 1.03i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (2.41 + 1.98i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-4.96 + 2.05i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.321 + 3.26i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (-4.07 + 0.809i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (0.827 - 2.72i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (3.88 + 3.88i)T + 31iT^{2} \)
37 \( 1 + (2.61 + 0.257i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-1.20 - 6.07i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (9.65 + 5.16i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-2.67 - 6.45i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.861 + 2.84i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-5.35 + 4.39i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-5.71 - 10.6i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (-2.05 - 3.83i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (1.74 + 2.61i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-6.90 - 4.61i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.304 + 0.734i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (5.51 - 0.543i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (6.60 + 1.31i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-10.0 - 10.0i)T + 97iT^{2} \)
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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

−11.47127349786829451642956852774, −9.966708246337619024361651029886, −9.137438814524293562191020623186, −8.440491802546204230945941561425, −7.58173487341169722863446055702, −6.80585766163428859855840519580, −5.47949169287321000975468563421, −4.51603467112180805735002529884, −3.32816058742216865636440875617, −0.893711259457127491507841946860, 1.79266521355358861184362368634, 3.31662301241413692925169780592, 3.88932087137612894784119966176, 5.19895930402404071268901013245, 6.88867359809014541015714337304, 7.82950927137054673653614749451, 8.842114541443085767899189682803, 9.736814272604654114646880308413, 10.41481872618459933380417712740, 11.43802875090498761410333190186

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