Properties

Label 2-384-384.131-c1-0-58
Degree $2$
Conductor $384$
Sign $-0.987 + 0.154i$
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  + (0.0871 − 1.41i)2-s + (1.16 − 1.28i)3-s + (−1.98 − 0.246i)4-s + (−0.108 + 0.356i)5-s + (−1.70 − 1.75i)6-s + (−0.537 − 2.70i)7-s + (−0.520 + 2.78i)8-s + (−0.283 − 2.98i)9-s + (0.494 + 0.183i)10-s + (−2.68 + 0.264i)11-s + (−2.62 + 2.25i)12-s + (−0.608 − 2.00i)13-s + (−3.86 + 0.523i)14-s + (0.331 + 0.554i)15-s + (3.87 + 0.977i)16-s + (1.76 + 0.729i)17-s + ⋯
L(s)  = 1  + (0.0616 − 0.998i)2-s + (0.672 − 0.739i)3-s + (−0.992 − 0.123i)4-s + (−0.0484 + 0.159i)5-s + (−0.696 − 0.717i)6-s + (−0.203 − 1.02i)7-s + (−0.184 + 0.982i)8-s + (−0.0944 − 0.995i)9-s + (0.156 + 0.0581i)10-s + (−0.808 + 0.0796i)11-s + (−0.758 + 0.651i)12-s + (−0.168 − 0.556i)13-s + (−1.03 + 0.139i)14-s + (0.0854 + 0.143i)15-s + (0.969 + 0.244i)16-s + (0.426 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.987 + 0.154i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102505 - 1.31651i\)
\(L(\frac12)\) \(\approx\) \(0.102505 - 1.31651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 + 1.41i)T \)
3 \( 1 + (-1.16 + 1.28i)T \)
good5 \( 1 + (0.108 - 0.356i)T + (-4.15 - 2.77i)T^{2} \)
7 \( 1 + (0.537 + 2.70i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.68 - 0.264i)T + (10.7 - 2.14i)T^{2} \)
13 \( 1 + (0.608 + 2.00i)T + (-10.8 + 7.22i)T^{2} \)
17 \( 1 + (-1.76 - 0.729i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (-0.122 + 0.228i)T + (-10.5 - 15.7i)T^{2} \)
23 \( 1 + (3.72 + 5.57i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-4.75 - 0.468i)T + (28.4 + 5.65i)T^{2} \)
31 \( 1 + (0.807 + 0.807i)T + 31iT^{2} \)
37 \( 1 + (0.189 + 0.353i)T + (-20.5 + 30.7i)T^{2} \)
41 \( 1 + (-1.36 - 2.04i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-7.31 - 6.00i)T + (8.38 + 42.1i)T^{2} \)
47 \( 1 + (-7.57 - 3.13i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (2.90 - 0.285i)T + (51.9 - 10.3i)T^{2} \)
59 \( 1 + (-2.18 + 7.21i)T + (-49.0 - 32.7i)T^{2} \)
61 \( 1 + (2.97 - 2.43i)T + (11.9 - 59.8i)T^{2} \)
67 \( 1 + (-8.17 - 9.96i)T + (-13.0 + 65.7i)T^{2} \)
71 \( 1 + (3.16 + 15.9i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.28 + 0.254i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (5.30 + 12.8i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-3.97 + 7.42i)T + (-46.1 - 69.0i)T^{2} \)
89 \( 1 + (-7.46 - 4.98i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-7.19 + 7.19i)T - 97iT^{2} \)
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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

−10.71181786868394061442882605807, −10.24400766228856828209052680381, −9.163975141674636168860597495836, −8.107903128565392551658029564394, −7.44470084865596696926441551190, −6.11548770525063295084002148455, −4.62121978423951574075318000313, −3.42849049525108090943618800725, −2.47648109361525886854592489570, −0.829219271869648418983441344425, 2.63733551767642363298300347196, 3.95821814937567363128878528833, 5.10122482147412228357381872803, 5.81418518007852135199581499290, 7.24089957649519302208110375015, 8.188594672324711816892323270235, 8.898484220794309930318509704296, 9.630539290718890124458927389548, 10.50865318569521467042306608726, 11.96063776923233504838110126680

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