Properties

Label 2-384-384.107-c1-0-23
Degree $2$
Conductor $384$
Sign $-0.555 - 0.831i$
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  + (−0.201 + 1.39i)2-s + (1.68 + 0.390i)3-s + (−1.91 − 0.563i)4-s + (0.582 − 0.176i)5-s + (−0.886 + 2.28i)6-s + (−0.838 + 4.21i)7-s + (1.17 − 2.57i)8-s + (2.69 + 1.31i)9-s + (0.130 + 0.851i)10-s + (−0.134 + 1.37i)11-s + (−3.01 − 1.70i)12-s + (−3.09 − 0.939i)13-s + (−5.72 − 2.02i)14-s + (1.05 − 0.0705i)15-s + (3.36 + 2.16i)16-s + (6.20 − 2.56i)17-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (0.974 + 0.225i)3-s + (−0.959 − 0.281i)4-s + (0.260 − 0.0790i)5-s + (−0.361 + 0.932i)6-s + (−0.316 + 1.59i)7-s + (0.415 − 0.909i)8-s + (0.898 + 0.439i)9-s + (0.0411 + 0.269i)10-s + (−0.0406 + 0.413i)11-s + (−0.871 − 0.490i)12-s + (−0.859 − 0.260i)13-s + (−1.53 − 0.540i)14-s + (0.271 − 0.0182i)15-s + (0.841 + 0.540i)16-s + (1.50 − 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735028 + 1.37543i\)
\(L(\frac12)\) \(\approx\) \(0.735028 + 1.37543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.201 - 1.39i)T \)
3 \( 1 + (-1.68 - 0.390i)T \)
good5 \( 1 + (-0.582 + 0.176i)T + (4.15 - 2.77i)T^{2} \)
7 \( 1 + (0.838 - 4.21i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.134 - 1.37i)T + (-10.7 - 2.14i)T^{2} \)
13 \( 1 + (3.09 + 0.939i)T + (10.8 + 7.22i)T^{2} \)
17 \( 1 + (-6.20 + 2.56i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (1.30 - 0.696i)T + (10.5 - 15.7i)T^{2} \)
23 \( 1 + (2.82 - 4.22i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.282 - 2.86i)T + (-28.4 + 5.65i)T^{2} \)
31 \( 1 + (-4.60 + 4.60i)T - 31iT^{2} \)
37 \( 1 + (-0.0723 - 0.0386i)T + (20.5 + 30.7i)T^{2} \)
41 \( 1 + (-5.98 + 8.96i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (5.22 + 6.36i)T + (-8.38 + 42.1i)T^{2} \)
47 \( 1 + (-5.23 + 2.16i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-0.882 + 8.95i)T + (-51.9 - 10.3i)T^{2} \)
59 \( 1 + (-3.26 + 0.991i)T + (49.0 - 32.7i)T^{2} \)
61 \( 1 + (1.31 - 1.59i)T + (-11.9 - 59.8i)T^{2} \)
67 \( 1 + (-5.94 - 4.87i)T + (13.0 + 65.7i)T^{2} \)
71 \( 1 + (0.636 - 3.20i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-6.53 + 1.29i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-1.53 + 3.71i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-9.82 + 5.25i)T + (46.1 - 69.0i)T^{2} \)
89 \( 1 + (-1.61 + 1.07i)T + (34.0 - 82.2i)T^{2} \)
97 \( 1 + (11.7 + 11.7i)T + 97iT^{2} \)
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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

−11.99092400017333905024636581657, −10.01680506672090666280585422432, −9.673037299467227490766229775343, −8.862697109017723753752616485826, −7.958020740953588754247866541335, −7.19018868728270364095450356041, −5.75369183021853574198609341602, −5.13306100710728733695895631161, −3.60871046817146794628730886707, −2.21780965960092992912403235209, 1.08903322415987108253547724965, 2.62188171366337334002074360647, 3.71543812643930669073378846766, 4.53995977863681449553363317042, 6.39910753072686457048629770606, 7.69809095547159453281577283269, 8.180238961974508651538838326616, 9.578352604817757104027870515708, 10.03592137725428904951455209621, 10.73071779911553893286160557184

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