Properties

Label 2-384-128.93-c1-0-24
Degree $2$
Conductor $384$
Sign $0.404 + 0.914i$
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  + (1.40 − 0.153i)2-s + (−0.995 − 0.0980i)3-s + (1.95 − 0.431i)4-s + (−0.589 − 1.10i)5-s + (−1.41 + 0.0147i)6-s + (−0.582 − 2.93i)7-s + (2.67 − 0.905i)8-s + (0.980 + 0.195i)9-s + (−0.997 − 1.45i)10-s + (0.231 + 0.189i)11-s + (−1.98 + 0.237i)12-s + (−0.103 − 0.0553i)13-s + (−1.26 − 4.03i)14-s + (0.478 + 1.15i)15-s + (3.62 − 1.68i)16-s + (1.89 − 4.57i)17-s + ⋯
L(s)  = 1  + (0.994 − 0.108i)2-s + (−0.574 − 0.0565i)3-s + (0.976 − 0.215i)4-s + (−0.263 − 0.493i)5-s + (−0.577 + 0.00604i)6-s + (−0.220 − 1.10i)7-s + (0.947 − 0.320i)8-s + (0.326 + 0.0650i)9-s + (−0.315 − 0.461i)10-s + (0.0696 + 0.0571i)11-s + (−0.573 + 0.0685i)12-s + (−0.0287 − 0.0153i)13-s + (−0.339 − 1.07i)14-s + (0.123 + 0.298i)15-s + (0.907 − 0.420i)16-s + (0.459 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66584 - 1.08422i\)
\(L(\frac12)\) \(\approx\) \(1.66584 - 1.08422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.153i)T \)
3 \( 1 + (0.995 + 0.0980i)T \)
good5 \( 1 + (0.589 + 1.10i)T + (-2.77 + 4.15i)T^{2} \)
7 \( 1 + (0.582 + 2.93i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.231 - 0.189i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (0.103 + 0.0553i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (-1.89 + 4.57i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.01 + 0.308i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (7.01 - 4.69i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-6.73 - 8.20i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (-2.45 + 2.45i)T - 31iT^{2} \)
37 \( 1 + (2.39 - 7.88i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (-3.86 - 5.78i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.901 - 0.0888i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (5.26 + 2.18i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (5.74 - 7.00i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (2.09 - 1.12i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (-0.688 + 6.98i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (0.178 - 1.80i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (4.83 - 0.961i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (1.07 - 5.39i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (5.60 - 2.31i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.35 - 14.3i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (-4.55 - 3.04i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-11.4 + 11.4i)T - 97iT^{2} \)
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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

−11.46932473305350487108729776448, −10.40554689087872536994561881568, −9.759909557919955232804276001244, −8.048608654812695997681503229520, −7.15790557151360520383597476744, −6.31266883339981897329605798434, −5.08241115260620665933977818532, −4.36059373059672648308422517272, −3.16614613120634546715457446731, −1.15575221002053767457489241331, 2.21086770193785912934426994351, 3.50074431748618256447131555531, 4.65561527971335576736303052764, 5.91851626391059253550358237559, 6.27370012006222195685644532092, 7.54888429474470004588530010910, 8.554852455073094840465845963802, 10.05229627511056812521694448425, 10.77751049389350062744398139398, 11.87075198372347241862079962536

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