Properties

Label 2-384-128.93-c1-0-8
Degree $2$
Conductor $384$
Sign $0.320 - 0.947i$
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  + (1.27 + 0.610i)2-s + (−0.995 − 0.0980i)3-s + (1.25 + 1.55i)4-s + (−1.28 − 2.39i)5-s + (−1.20 − 0.732i)6-s + (0.997 + 5.01i)7-s + (0.649 + 2.75i)8-s + (0.980 + 0.195i)9-s + (−0.171 − 3.84i)10-s + (0.439 + 0.360i)11-s + (−1.09 − 1.67i)12-s + (3.44 + 1.84i)13-s + (−1.78 + 7.00i)14-s + (1.04 + 2.51i)15-s + (−0.852 + 3.90i)16-s + (−0.200 + 0.484i)17-s + ⋯
L(s)  = 1  + (0.902 + 0.431i)2-s + (−0.574 − 0.0565i)3-s + (0.627 + 0.778i)4-s + (−0.573 − 1.07i)5-s + (−0.493 − 0.299i)6-s + (0.377 + 1.89i)7-s + (0.229 + 0.973i)8-s + (0.326 + 0.0650i)9-s + (−0.0540 − 1.21i)10-s + (0.132 + 0.108i)11-s + (−0.316 − 0.482i)12-s + (0.955 + 0.510i)13-s + (−0.478 + 1.87i)14-s + (0.268 + 0.648i)15-s + (−0.213 + 0.977i)16-s + (−0.0486 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.320 - 0.947i$
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: \((2,\ 384,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50720 + 1.08080i\)
\(L(\frac12)\) \(\approx\) \(1.50720 + 1.08080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.610i)T \)
3 \( 1 + (0.995 + 0.0980i)T \)
good5 \( 1 + (1.28 + 2.39i)T + (-2.77 + 4.15i)T^{2} \)
7 \( 1 + (-0.997 - 5.01i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.439 - 0.360i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (-3.44 - 1.84i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (0.200 - 0.484i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-4.89 + 1.48i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (4.43 - 2.96i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (2.77 + 3.38i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (-3.56 + 3.56i)T - 31iT^{2} \)
37 \( 1 + (-0.893 + 2.94i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (5.07 + 7.59i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (10.6 - 1.04i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (-4.81 - 1.99i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.501 - 0.611i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (-9.69 + 5.18i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (0.0767 - 0.779i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (-1.28 + 13.0i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (-10.4 + 2.07i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.349 - 1.75i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-8.80 + 3.64i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-1.74 - 5.75i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (1.79 + 1.19i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-8.59 + 8.59i)T - 97iT^{2} \)
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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

−11.85044860391393203942013079958, −11.26246798753519741940910846664, −9.424332034856795067178303019610, −8.541850868228925797890905045919, −7.85141598912988524463435969213, −6.40089429239863334847404839266, −5.55423316848137906959784242143, −4.92071855133622355262289970290, −3.73421344645439339876065286595, −1.98461481746937226945969070783, 1.13918643783380467577637526202, 3.33383270397556226040640179437, 3.93346355475211117598521195197, 5.10108574788155620696142257065, 6.46694963827727388598605978556, 7.05675344918386223220811426980, 7.995453389915012997827849901125, 10.18442233399103791532519928346, 10.34606485291413197298368363123, 11.31789836859088798799969169715

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