Properties

Label 2-384-384.323-c1-0-19
Degree $2$
Conductor $384$
Sign $0.542 - 0.840i$
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  + (−0.0211 + 1.41i)2-s + (0.382 − 1.68i)3-s + (−1.99 − 0.0598i)4-s + (2.03 + 0.618i)5-s + (2.38 + 0.577i)6-s + (0.647 + 3.25i)7-s + (0.126 − 2.82i)8-s + (−2.70 − 1.29i)9-s + (−0.918 + 2.87i)10-s + (0.158 + 1.61i)11-s + (−0.866 + 3.35i)12-s + (3.48 − 1.05i)13-s + (−4.61 + 0.847i)14-s + (1.82 − 3.20i)15-s + (3.99 + 0.239i)16-s + (1.96 + 0.813i)17-s + ⋯
L(s)  = 1  + (−0.0149 + 0.999i)2-s + (0.221 − 0.975i)3-s + (−0.999 − 0.0299i)4-s + (0.912 + 0.276i)5-s + (0.971 + 0.235i)6-s + (0.244 + 1.23i)7-s + (0.0448 − 0.998i)8-s + (−0.902 − 0.431i)9-s + (−0.290 + 0.908i)10-s + (0.0478 + 0.485i)11-s + (−0.250 + 0.968i)12-s + (0.967 − 0.293i)13-s + (−1.23 + 0.226i)14-s + (0.471 − 0.828i)15-s + (0.998 + 0.0597i)16-s + (0.476 + 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.542 - 0.840i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36720 + 0.744898i\)
\(L(\frac12)\) \(\approx\) \(1.36720 + 0.744898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0211 - 1.41i)T \)
3 \( 1 + (-0.382 + 1.68i)T \)
good5 \( 1 + (-2.03 - 0.618i)T + (4.15 + 2.77i)T^{2} \)
7 \( 1 + (-0.647 - 3.25i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.158 - 1.61i)T + (-10.7 + 2.14i)T^{2} \)
13 \( 1 + (-3.48 + 1.05i)T + (10.8 - 7.22i)T^{2} \)
17 \( 1 + (-1.96 - 0.813i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (-3.61 - 1.93i)T + (10.5 + 15.7i)T^{2} \)
23 \( 1 + (1.30 + 1.95i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-0.175 + 1.77i)T + (-28.4 - 5.65i)T^{2} \)
31 \( 1 + (-7.41 - 7.41i)T + 31iT^{2} \)
37 \( 1 + (-4.45 + 2.38i)T + (20.5 - 30.7i)T^{2} \)
41 \( 1 + (6.01 + 9.00i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (5.84 - 7.12i)T + (-8.38 - 42.1i)T^{2} \)
47 \( 1 + (7.55 + 3.12i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-0.0669 - 0.679i)T + (-51.9 + 10.3i)T^{2} \)
59 \( 1 + (6.68 + 2.02i)T + (49.0 + 32.7i)T^{2} \)
61 \( 1 + (5.91 + 7.21i)T + (-11.9 + 59.8i)T^{2} \)
67 \( 1 + (0.592 - 0.486i)T + (13.0 - 65.7i)T^{2} \)
71 \( 1 + (1.70 + 8.56i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-0.735 - 0.146i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (6.16 + 14.8i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (0.488 + 0.260i)T + (46.1 + 69.0i)T^{2} \)
89 \( 1 + (-9.97 - 6.66i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (2.86 - 2.86i)T - 97iT^{2} \)
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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

−11.85559249948534985867531405703, −10.29119760577328730408658267420, −9.346953799326690522990831481337, −8.482374486867790654557843904630, −7.81899502332451167377464851917, −6.51128850832439198058834644722, −6.02185379278205395239673500216, −5.13163836416092538124969689577, −3.17502244587479699827136918314, −1.65874700558221295291735804003, 1.28242523951838294295916162364, 3.02954548650837829361478325741, 4.02995729363201704428584780750, 4.98465024872901574381362583733, 6.04190189305157321706322455827, 7.86148065809986962576319915001, 8.766879247394191439378671621983, 9.810524480174469041666651799175, 10.05391778176233296508487155102, 11.19010302549774752140748680015

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