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Properties

Label 2-384-128.101-c1-0-3
Degree $2$
Conductor $384$
Sign $-0.881 - 0.472i$
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  + (−1.20 + 0.743i)2-s + (−0.881 + 0.471i)3-s + (0.893 − 1.78i)4-s + (2.29 + 2.79i)5-s + (0.710 − 1.22i)6-s + (−2.80 + 1.87i)7-s + (0.255 + 2.81i)8-s + (0.555 − 0.831i)9-s + (−4.82 − 1.65i)10-s + (2.03 − 0.618i)11-s + (0.0551 + 1.99i)12-s + (−0.229 − 0.188i)13-s + (1.98 − 4.34i)14-s + (−3.33 − 1.38i)15-s + (−2.40 − 3.19i)16-s + (2.17 − 0.901i)17-s + ⋯
L(s)  = 1  + (−0.850 + 0.525i)2-s + (−0.509 + 0.272i)3-s + (0.446 − 0.894i)4-s + (1.02 + 1.24i)5-s + (0.289 − 0.499i)6-s + (−1.06 + 0.708i)7-s + (0.0903 + 0.995i)8-s + (0.185 − 0.277i)9-s + (−1.52 − 0.522i)10-s + (0.614 − 0.186i)11-s + (0.0159 + 0.577i)12-s + (−0.0636 − 0.0522i)13-s + (0.529 − 1.16i)14-s + (−0.861 − 0.356i)15-s + (−0.600 − 0.799i)16-s + (0.528 − 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.881 - 0.472i$
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: \((2,\ 384,\ (\ :1/2),\ -0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170151 + 0.676936i\)
\(L(\frac12)\) \(\approx\) \(0.170151 + 0.676936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.743i)T \)
3 \( 1 + (0.881 - 0.471i)T \)
good5 \( 1 + (-2.29 - 2.79i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (2.80 - 1.87i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-2.03 + 0.618i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (0.229 + 0.188i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-2.17 + 0.901i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.706 - 7.16i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (7.67 - 1.52i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.45 + 4.79i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (1.34 + 1.34i)T + 31iT^{2} \)
37 \( 1 + (11.2 + 1.10i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-1.95 - 9.84i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.88 + 2.07i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (0.144 + 0.349i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-2.39 - 7.89i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (-5.66 + 4.64i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (1.29 + 2.43i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (0.766 + 1.43i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-7.40 - 11.0i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-8.54 - 5.70i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-6.07 + 14.6i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-11.9 + 1.17i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (0.238 + 0.0474i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-0.798 - 0.798i)T + 97iT^{2} \)
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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

−11.53007521738040147823196420320, −10.30099271310221235450435889238, −9.972210567695078234283186962153, −9.322044117631111425833135888334, −7.971604530639508815552506243375, −6.73734002969690887725215942072, −6.09194440644320198561078623486, −5.63366767586685625887059120101, −3.42908285732323935349283612469, −2.00770190992439184515878628725, 0.64024487587796594444162599493, 1.95001120495676682028486470965, 3.71176443084294518533304429788, 5.09301700285202490267635361856, 6.40810883551549321548699446701, 7.10188853699921451026930281709, 8.501452290786522381903866423206, 9.290006657855265540821124059813, 9.968395410455612695957742387717, 10.69409935708217250183469885153

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