Properties

Label 2-384-12.11-c3-0-41
Degree $2$
Conductor $384$
Sign $-0.585 + 0.810i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.04 − 4.21i)3-s + 9.33i·5-s − 36.3i·7-s + (−8.50 − 25.6i)9-s + 48.4·11-s − 25.8·13-s + (39.3 + 28.3i)15-s + 74.2i·17-s − 82.9i·19-s + (−153. − 110. i)21-s − 179.·23-s + 37.8·25-s + (−133. − 42.1i)27-s − 122. i·29-s + 64.1i·31-s + ⋯
L(s)  = 1  + (0.585 − 0.810i)3-s + 0.834i·5-s − 1.96i·7-s + (−0.314 − 0.949i)9-s + 1.32·11-s − 0.552·13-s + (0.676 + 0.488i)15-s + 1.05i·17-s − 1.00i·19-s + (−1.59 − 1.14i)21-s − 1.62·23-s + 0.303·25-s + (−0.953 − 0.300i)27-s − 0.783i·29-s + 0.371i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.585 + 0.810i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.585 + 0.810i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.964449867\)
\(L(\frac12)\) \(\approx\) \(1.964449867\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-3.04 + 4.21i)T \)
good5 \( 1 - 9.33iT - 125T^{2} \)
7 \( 1 + 36.3iT - 343T^{2} \)
11 \( 1 - 48.4T + 1.33e3T^{2} \)
13 \( 1 + 25.8T + 2.19e3T^{2} \)
17 \( 1 - 74.2iT - 4.91e3T^{2} \)
19 \( 1 + 82.9iT - 6.85e3T^{2} \)
23 \( 1 + 179.T + 1.21e4T^{2} \)
29 \( 1 + 122. iT - 2.43e4T^{2} \)
31 \( 1 - 64.1iT - 2.97e4T^{2} \)
37 \( 1 + 5.01T + 5.06e4T^{2} \)
41 \( 1 + 325. iT - 6.89e4T^{2} \)
43 \( 1 + 321. iT - 7.95e4T^{2} \)
47 \( 1 - 95.9T + 1.03e5T^{2} \)
53 \( 1 + 185. iT - 1.48e5T^{2} \)
59 \( 1 - 226.T + 2.05e5T^{2} \)
61 \( 1 + 198.T + 2.26e5T^{2} \)
67 \( 1 - 23.9iT - 3.00e5T^{2} \)
71 \( 1 + 399.T + 3.57e5T^{2} \)
73 \( 1 - 669.T + 3.89e5T^{2} \)
79 \( 1 - 229. iT - 4.93e5T^{2} \)
83 \( 1 + 321.T + 5.71e5T^{2} \)
89 \( 1 + 131. iT - 7.04e5T^{2} \)
97 \( 1 - 136.T + 9.12e5T^{2} \)
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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

−10.54199780926122556047283308017, −9.804844269128418079768251932381, −8.601212817862206700328395267950, −7.50729004828887650324535431393, −6.95831159557778159031241367716, −6.26840509789635564963212274279, −4.18069425280876860968356637074, −3.51948264422411573424879881868, −1.95597374755511047768285280581, −0.61112346466220863338819025837, 1.82509745727643049711576726172, 3.01503481128310974519102358630, 4.37932015157603113377322931697, 5.28155636790816457979099517273, 6.19125125062164742160380001015, 7.907091302254793671473653430487, 8.727555518413275360206722841966, 9.314005412705479581671761079095, 9.889267186536616564189038562159, 11.48788204703634052825313711600

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