Properties

Label 2-384-128.85-c1-0-13
Degree $2$
Conductor $384$
Sign $0.978 - 0.207i$
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  + (−1.34 + 0.441i)2-s + (−0.634 + 0.773i)3-s + (1.61 − 1.18i)4-s + (2.46 − 0.748i)5-s + (0.511 − 1.31i)6-s + (2.57 + 0.512i)7-s + (−1.64 + 2.30i)8-s + (−0.195 − 0.980i)9-s + (−2.98 + 2.09i)10-s + (−0.342 − 0.0337i)11-s + (−0.104 + 1.99i)12-s + (1.04 − 3.44i)13-s + (−3.68 + 0.448i)14-s + (−0.987 + 2.38i)15-s + (1.18 − 3.81i)16-s + (−2.21 − 5.34i)17-s + ⋯
L(s)  = 1  + (−0.950 + 0.312i)2-s + (−0.366 + 0.446i)3-s + (0.805 − 0.592i)4-s + (1.10 − 0.334i)5-s + (0.208 − 0.538i)6-s + (0.973 + 0.193i)7-s + (−0.579 + 0.814i)8-s + (−0.0650 − 0.326i)9-s + (−0.944 + 0.662i)10-s + (−0.103 − 0.0101i)11-s + (−0.0302 + 0.576i)12-s + (0.289 − 0.955i)13-s + (−0.984 + 0.119i)14-s + (−0.254 + 0.615i)15-s + (0.296 − 0.954i)16-s + (−0.537 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.978 - 0.207i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05283 + 0.110453i\)
\(L(\frac12)\) \(\approx\) \(1.05283 + 0.110453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.441i)T \)
3 \( 1 + (0.634 - 0.773i)T \)
good5 \( 1 + (-2.46 + 0.748i)T + (4.15 - 2.77i)T^{2} \)
7 \( 1 + (-2.57 - 0.512i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.342 + 0.0337i)T + (10.7 + 2.14i)T^{2} \)
13 \( 1 + (-1.04 + 3.44i)T + (-10.8 - 7.22i)T^{2} \)
17 \( 1 + (2.21 + 5.34i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (-3.09 + 1.65i)T + (10.5 - 15.7i)T^{2} \)
23 \( 1 + (1.10 - 1.65i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.640 - 6.50i)T + (-28.4 + 5.65i)T^{2} \)
31 \( 1 + (-6.17 - 6.17i)T + 31iT^{2} \)
37 \( 1 + (-3.27 + 6.12i)T + (-20.5 - 30.7i)T^{2} \)
41 \( 1 + (-3.68 - 2.46i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-0.676 - 0.824i)T + (-8.38 + 42.1i)T^{2} \)
47 \( 1 + (7.18 - 2.97i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-0.897 + 9.11i)T + (-51.9 - 10.3i)T^{2} \)
59 \( 1 + (-1.35 - 4.48i)T + (-49.0 + 32.7i)T^{2} \)
61 \( 1 + (-8.45 - 6.93i)T + (11.9 + 59.8i)T^{2} \)
67 \( 1 + (0.239 + 0.196i)T + (13.0 + 65.7i)T^{2} \)
71 \( 1 + (-1.57 + 7.89i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (11.8 - 2.35i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (7.38 + 3.05i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-4.94 - 9.25i)T + (-46.1 + 69.0i)T^{2} \)
89 \( 1 + (-3.47 - 5.19i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (-9.98 - 9.98i)T + 97iT^{2} \)
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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

−11.14517101943390543975476223662, −10.30740382471189190394599673374, −9.504684948757820189532948333906, −8.794109069915389709177533486132, −7.78925197410346318376938030355, −6.64379804849421522195066842471, −5.45924185817723365701524695880, −5.03948337801573794290157538682, −2.72744148576115340470859383120, −1.21932286035071962708908018495, 1.48130741955482623942577691277, 2.35629770100016835186186249086, 4.27067529088495844890841970960, 5.95706515691575199602019428683, 6.53329604225743920279077282820, 7.77037284853577083861786340944, 8.476482174501109382528841842355, 9.679315715782574062040270075219, 10.32029475556335761849313903567, 11.29419719186517359039835872776

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