Properties

Label 2-384-384.227-c1-0-29
Degree $2$
Conductor $384$
Sign $0.215 - 0.976i$
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  + (−0.608 + 1.27i)2-s + (1.73 − 0.00196i)3-s + (−1.25 − 1.55i)4-s + (1.34 + 2.51i)5-s + (−1.05 + 2.21i)6-s + (1.32 − 0.263i)7-s + (2.74 − 0.661i)8-s + (2.99 − 0.00679i)9-s + (−4.02 + 0.185i)10-s + (0.564 − 0.688i)11-s + (−2.18 − 2.68i)12-s + (2.06 − 3.86i)13-s + (−0.469 + 1.85i)14-s + (2.33 + 4.35i)15-s + (−0.829 + 3.91i)16-s + (−1.46 − 0.606i)17-s + ⋯
L(s)  = 1  + (−0.430 + 0.902i)2-s + (0.999 − 0.00113i)3-s + (−0.629 − 0.776i)4-s + (0.601 + 1.12i)5-s + (−0.429 + 0.903i)6-s + (0.500 − 0.0995i)7-s + (0.972 − 0.233i)8-s + (0.999 − 0.00226i)9-s + (−1.27 + 0.0585i)10-s + (0.170 − 0.207i)11-s + (−0.630 − 0.776i)12-s + (0.572 − 1.07i)13-s + (−0.125 + 0.494i)14-s + (0.602 + 1.12i)15-s + (−0.207 + 0.978i)16-s + (−0.355 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.215 - 0.976i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.215 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30196 + 1.04614i\)
\(L(\frac12)\) \(\approx\) \(1.30196 + 1.04614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 - 1.27i)T \)
3 \( 1 + (-1.73 + 0.00196i)T \)
good5 \( 1 + (-1.34 - 2.51i)T + (-2.77 + 4.15i)T^{2} \)
7 \( 1 + (-1.32 + 0.263i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.564 + 0.688i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (-2.06 + 3.86i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (1.46 + 0.606i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (7.79 - 2.36i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (1.85 - 1.23i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.46 - 1.78i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (-3.09 - 3.09i)T + 31iT^{2} \)
37 \( 1 + (-7.34 - 2.22i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (6.44 - 4.30i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (0.929 - 0.0915i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (12.4 + 5.15i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-5.74 + 7.00i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (6.05 + 11.3i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (6.35 + 0.625i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (-0.311 + 3.16i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (-11.5 + 2.29i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.430 - 2.16i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (1.16 + 2.81i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (4.25 - 1.29i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-2.36 + 3.54i)T + (-34.0 - 82.2i)T^{2} \)
97 \( 1 + (-1.97 + 1.97i)T - 97iT^{2} \)
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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

−11.05501106759073706467250955072, −10.33708585672773359101056148386, −9.699171851790733723022838527551, −8.362705925722451822635185181803, −8.115520105027029619977273843937, −6.76835871925046632504836383284, −6.22142286454455042095439463829, −4.72600635368081542830350457733, −3.33870613058417579559594059074, −1.84308874083688813037148845890, 1.50581668330119603849443167523, 2.36023717598070921589644138516, 4.14151760213258782473744884768, 4.65640622509886458624272066436, 6.50302905560163706850007755446, 7.969006160742398961036090914796, 8.710272290658170227822555321508, 9.137537409595863138754527176451, 10.03335990270359976474641153554, 11.07647243126657346313138827782

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