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Properties

Label 2-384-128.101-c1-0-17
Degree $2$
Conductor $384$
Sign $0.876 - 0.480i$
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  + (1.38 + 0.283i)2-s + (−0.881 + 0.471i)3-s + (1.83 + 0.785i)4-s + (−0.311 − 0.379i)5-s + (−1.35 + 0.403i)6-s + (2.05 − 1.37i)7-s + (2.32 + 1.60i)8-s + (0.555 − 0.831i)9-s + (−0.324 − 0.614i)10-s + (0.708 − 0.215i)11-s + (−1.99 + 0.174i)12-s + (1.87 + 1.54i)13-s + (3.24 − 1.32i)14-s + (0.453 + 0.187i)15-s + (2.76 + 2.88i)16-s + (0.338 − 0.140i)17-s + ⋯
L(s)  = 1  + (0.979 + 0.200i)2-s + (−0.509 + 0.272i)3-s + (0.919 + 0.392i)4-s + (−0.139 − 0.169i)5-s + (−0.553 + 0.164i)6-s + (0.777 − 0.519i)7-s + (0.822 + 0.569i)8-s + (0.185 − 0.277i)9-s + (−0.102 − 0.194i)10-s + (0.213 − 0.0648i)11-s + (−0.575 + 0.0502i)12-s + (0.521 + 0.427i)13-s + (0.866 − 0.353i)14-s + (0.117 + 0.0485i)15-s + (0.691 + 0.722i)16-s + (0.0822 − 0.0340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.876 - 0.480i$
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: \((2,\ 384,\ (\ :1/2),\ 0.876 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18586 + 0.559941i\)
\(L(\frac12)\) \(\approx\) \(2.18586 + 0.559941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.283i)T \)
3 \( 1 + (0.881 - 0.471i)T \)
good5 \( 1 + (0.311 + 0.379i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-2.05 + 1.37i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-0.708 + 0.215i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-1.87 - 1.54i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-0.338 + 0.140i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.634 - 6.44i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (7.28 - 1.44i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.561 + 1.84i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (5.57 + 5.57i)T + 31iT^{2} \)
37 \( 1 + (-6.92 - 0.681i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (0.989 + 4.97i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (5.69 + 3.04i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (1.01 + 2.44i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (2.92 + 9.62i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (8.85 - 7.26i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-0.238 - 0.445i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (-1.29 - 2.41i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-0.719 - 1.07i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (0.675 + 0.451i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-3.41 + 8.25i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (11.2 - 1.11i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (9.94 + 1.97i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-1.02 - 1.02i)T + 97iT^{2} \)
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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

−11.64378136740116937443839286396, −10.75877595514292558268105182763, −9.885987810078444797782193560804, −8.302989762358513271469142061791, −7.59467055968497960718331787413, −6.34003483580751085176651148794, −5.59364424426620199676162935857, −4.36731414447397830681258373694, −3.77687349066173742865966072683, −1.77710229957004080504139516424, 1.60411158065659281031962977796, 3.05127339966517342619182212923, 4.47712683045955537079779590812, 5.35538030299224001002000704031, 6.27853083913904462674373850411, 7.26137764927665780460418201697, 8.285875591158980316965877695559, 9.646090024940872781561859601972, 10.97597744798518270507786180914, 11.22290589663022689618589628469

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