Properties

Label 2-384-128.101-c1-0-27
Degree $2$
Conductor $384$
Sign $0.545 + 0.838i$
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  + (1.41 − 0.0470i)2-s + (0.881 − 0.471i)3-s + (1.99 − 0.132i)4-s + (−2.41 − 2.94i)5-s + (1.22 − 0.707i)6-s + (−0.253 + 0.169i)7-s + (2.81 − 0.281i)8-s + (0.555 − 0.831i)9-s + (−3.55 − 4.04i)10-s + (−0.265 + 0.0805i)11-s + (1.69 − 1.05i)12-s + (1.32 + 1.08i)13-s + (−0.350 + 0.251i)14-s + (−3.52 − 1.45i)15-s + (3.96 − 0.530i)16-s + (4.25 − 1.76i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0332i)2-s + (0.509 − 0.272i)3-s + (0.997 − 0.0664i)4-s + (−1.08 − 1.31i)5-s + (0.499 − 0.288i)6-s + (−0.0958 + 0.0640i)7-s + (0.995 − 0.0996i)8-s + (0.185 − 0.277i)9-s + (−1.12 − 1.28i)10-s + (−0.0800 + 0.0242i)11-s + (0.489 − 0.305i)12-s + (0.366 + 0.300i)13-s + (−0.0936 + 0.0671i)14-s + (−0.908 − 0.376i)15-s + (0.991 − 0.132i)16-s + (1.03 − 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20624 - 1.19712i\)
\(L(\frac12)\) \(\approx\) \(2.20624 - 1.19712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0470i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (2.41 + 2.94i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (0.253 - 0.169i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.265 - 0.0805i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-1.32 - 1.08i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-4.25 + 1.76i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.116 - 1.17i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (7.69 - 1.52i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.259 + 0.854i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-7.54 - 7.54i)T + 31iT^{2} \)
37 \( 1 + (11.1 + 1.10i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-0.635 - 3.19i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-6.03 - 3.22i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-3.17 - 7.67i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.522 + 1.72i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-1.79 + 1.47i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (3.97 + 7.44i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (1.29 + 2.41i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (4.04 + 6.05i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (3.57 + 2.38i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-2.38 + 5.75i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-9.70 + 0.955i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (6.03 + 1.20i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-7.57 - 7.57i)T + 97iT^{2} \)
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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

−11.78158563518892914938961776635, −10.43491578501360449531387633947, −9.227541400835066777121189924307, −8.097997627400999438301904858373, −7.63972131377295712512071490867, −6.28312891655544750549127865717, −5.08871864097803301187313216425, −4.17254443465159067776751226258, −3.22636255438627287010467720271, −1.41613520679695779163136000148, 2.48485383589087831771952561546, 3.53654100886983553186899873336, 4.11834641351778206750447739750, 5.69289409129356547319840108097, 6.75459060177965203526720472606, 7.63497036445739824382269804381, 8.313920726212771880038790460723, 10.20846602060616128683058979632, 10.52987584388842317803822080427, 11.69894080524760524536313677990

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