Properties

Label 2-384-128.101-c1-0-24
Degree $2$
Conductor $384$
Sign $-0.493 + 0.869i$
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  + (−0.143 − 1.40i)2-s + (0.881 − 0.471i)3-s + (−1.95 + 0.403i)4-s + (−0.269 − 0.328i)5-s + (−0.789 − 1.17i)6-s + (3.15 − 2.11i)7-s + (0.848 + 2.69i)8-s + (0.555 − 0.831i)9-s + (−0.424 + 0.426i)10-s + (3.14 − 0.953i)11-s + (−1.53 + 1.27i)12-s + (−1.62 − 1.32i)13-s + (−3.42 − 4.14i)14-s + (−0.393 − 0.162i)15-s + (3.67 − 1.58i)16-s + (−2.65 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.101 − 0.994i)2-s + (0.509 − 0.272i)3-s + (−0.979 + 0.201i)4-s + (−0.120 − 0.147i)5-s + (−0.322 − 0.478i)6-s + (1.19 − 0.797i)7-s + (0.299 + 0.953i)8-s + (0.185 − 0.277i)9-s + (−0.134 + 0.134i)10-s + (0.947 − 0.287i)11-s + (−0.443 + 0.369i)12-s + (−0.449 − 0.368i)13-s + (−0.914 − 1.10i)14-s + (−0.101 − 0.0420i)15-s + (0.918 − 0.395i)16-s + (−0.644 + 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.493 + 0.869i$
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: \((2,\ 384,\ (\ :1/2),\ -0.493 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750680 - 1.28962i\)
\(L(\frac12)\) \(\approx\) \(0.750680 - 1.28962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.143 + 1.40i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (0.269 + 0.328i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-3.15 + 2.11i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-3.14 + 0.953i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (1.62 + 1.32i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (2.65 - 1.10i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.0395 + 0.401i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (9.04 - 1.79i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.40 + 4.63i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-0.112 - 0.112i)T + 31iT^{2} \)
37 \( 1 + (-8.95 - 0.881i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-2.48 - 12.5i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-2.40 - 1.28i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-2.88 - 6.97i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (1.22 + 4.03i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (2.95 - 2.42i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (2.79 + 5.23i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (-6.18 - 11.5i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (4.93 + 7.38i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-8.60 - 5.75i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (3.45 - 8.33i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.02 + 0.297i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-5.02 - 0.999i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-7.81 - 7.81i)T + 97iT^{2} \)
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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

−11.20025019212848374212877220945, −10.14902414081096516764847225299, −9.344305803535407585128392509018, −8.100194124100700949968668453672, −7.898157357720757303035012615868, −6.22606269091666532265508246883, −4.55926912522050051281282820199, −4.00665809278758155259572169987, −2.41573467579296256735487234417, −1.12850898429451474785151941234, 1.99941569396458402414880929970, 3.96068407121167068335755174391, 4.80919435951173519766142282979, 5.90506536646160294333337108177, 7.09274981719465721754728118935, 7.940068487243220953139829111050, 8.856419386932301398703296644213, 9.364129823233212030930143528484, 10.56304552445921057930352944180, 11.71831941566731126560040195296

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