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Properties

Label 2-384-128.29-c1-0-18
Degree $2$
Conductor $384$
Sign $0.0871 + 0.996i$
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  + (−0.0869 − 1.41i)2-s + (−0.0980 + 0.995i)3-s + (−1.98 + 0.245i)4-s + (−0.678 + 0.362i)5-s + (1.41 + 0.0517i)6-s + (−0.110 − 0.554i)7-s + (0.519 + 2.78i)8-s + (−0.980 − 0.195i)9-s + (0.570 + 0.925i)10-s + (3.39 − 4.14i)11-s + (−0.0498 − 1.99i)12-s + (2.37 − 4.44i)13-s + (−0.772 + 0.203i)14-s + (−0.294 − 0.710i)15-s + (3.87 − 0.974i)16-s + (1.50 − 3.62i)17-s + ⋯
L(s)  = 1  + (−0.0615 − 0.998i)2-s + (−0.0565 + 0.574i)3-s + (−0.992 + 0.122i)4-s + (−0.303 + 0.162i)5-s + (0.576 + 0.0211i)6-s + (−0.0416 − 0.209i)7-s + (0.183 + 0.982i)8-s + (−0.326 − 0.0650i)9-s + (0.180 + 0.292i)10-s + (1.02 − 1.24i)11-s + (−0.0143 − 0.577i)12-s + (0.658 − 1.23i)13-s + (−0.206 + 0.0544i)14-s + (−0.0759 − 0.183i)15-s + (0.969 − 0.243i)16-s + (0.364 − 0.879i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.0871 + 0.996i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.0871 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833521 - 0.763771i\)
\(L(\frac12)\) \(\approx\) \(0.833521 - 0.763771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0869 + 1.41i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (0.678 - 0.362i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.110 + 0.554i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-3.39 + 4.14i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (-2.37 + 4.44i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (-1.50 + 3.62i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.06 - 3.51i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (-6.43 + 4.30i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.104 + 0.0860i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (0.250 - 0.250i)T - 31iT^{2} \)
37 \( 1 + (8.81 + 2.67i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (-0.174 - 0.261i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-0.800 - 8.12i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (-7.48 - 3.09i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (5.71 + 4.69i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (2.98 + 5.59i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (14.5 + 1.43i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (-0.526 - 0.0518i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (4.44 - 0.884i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (1.65 - 8.32i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-12.2 + 5.08i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-5.70 + 1.72i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-12.2 - 8.17i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (0.824 - 0.824i)T - 97iT^{2} \)
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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

−11.00313241146546443660877574781, −10.50269146932429175808169548449, −9.361930034327432459702882547237, −8.688018343775711047044385594865, −7.67865394858825078054650952654, −6.04923575978024519794065997212, −5.03128762451632661254637411058, −3.65331831917714329259398665083, −3.15116992052282584030291173419, −0.922519610843337655692101463139, 1.52859260809021068315776388915, 3.79573153761034412214418709197, 4.79071418400540258228618607186, 6.07739054882260625586267840437, 6.91601530373169951836157659531, 7.54784731980035431299990725574, 8.903027679570675380089361245885, 9.173611489090432527310412374211, 10.55230894200866841838289589589, 11.90356276995424375900660072466

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