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Properties

Label 2-384-128.13-c1-0-31
Degree $2$
Conductor $384$
Sign $-0.975 + 0.220i$
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  + (0.915 − 1.07i)2-s + (−0.290 − 0.956i)3-s + (−0.322 − 1.97i)4-s + (−0.270 − 2.74i)5-s + (−1.29 − 0.563i)6-s + (−1.85 + 2.77i)7-s + (−2.42 − 1.46i)8-s + (−0.831 + 0.555i)9-s + (−3.20 − 2.22i)10-s + (4.99 − 2.66i)11-s + (−1.79 + 0.881i)12-s + (−3.79 − 0.374i)13-s + (1.29 + 4.53i)14-s + (−2.54 + 1.05i)15-s + (−3.79 + 1.27i)16-s + (−0.856 − 0.354i)17-s + ⋯
L(s)  = 1  + (0.647 − 0.761i)2-s + (−0.167 − 0.552i)3-s + (−0.161 − 0.986i)4-s + (−0.120 − 1.22i)5-s + (−0.529 − 0.230i)6-s + (−0.700 + 1.04i)7-s + (−0.856 − 0.516i)8-s + (−0.277 + 0.185i)9-s + (−1.01 − 0.702i)10-s + (1.50 − 0.804i)11-s + (−0.518 + 0.254i)12-s + (−1.05 − 0.103i)13-s + (0.345 + 1.21i)14-s + (−0.657 + 0.272i)15-s + (−0.948 + 0.317i)16-s + (−0.207 − 0.0860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.975 + 0.220i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162146 - 1.45160i\)
\(L(\frac12)\) \(\approx\) \(0.162146 - 1.45160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.915 + 1.07i)T \)
3 \( 1 + (0.290 + 0.956i)T \)
good5 \( 1 + (0.270 + 2.74i)T + (-4.90 + 0.975i)T^{2} \)
7 \( 1 + (1.85 - 2.77i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-4.99 + 2.66i)T + (6.11 - 9.14i)T^{2} \)
13 \( 1 + (3.79 + 0.374i)T + (12.7 + 2.53i)T^{2} \)
17 \( 1 + (0.856 + 0.354i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (-2.48 + 2.03i)T + (3.70 - 18.6i)T^{2} \)
23 \( 1 + (1.11 - 5.59i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-2.13 + 3.99i)T + (-16.1 - 24.1i)T^{2} \)
31 \( 1 + (-6.37 + 6.37i)T - 31iT^{2} \)
37 \( 1 + (5.01 - 6.11i)T + (-7.21 - 36.2i)T^{2} \)
41 \( 1 + (0.359 + 0.0715i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-3.25 + 10.7i)T + (-35.7 - 23.8i)T^{2} \)
47 \( 1 + (-3.33 + 8.04i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (0.428 + 0.802i)T + (-29.4 + 44.0i)T^{2} \)
59 \( 1 + (-0.295 + 0.0291i)T + (57.8 - 11.5i)T^{2} \)
61 \( 1 + (-10.5 + 3.19i)T + (50.7 - 33.8i)T^{2} \)
67 \( 1 + (7.60 - 2.30i)T + (55.7 - 37.2i)T^{2} \)
71 \( 1 + (-9.07 - 6.06i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (0.960 + 1.43i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-3.98 - 9.62i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-9.68 - 11.8i)T + (-16.1 + 81.4i)T^{2} \)
89 \( 1 + (0.373 + 1.87i)T + (-82.2 + 34.0i)T^{2} \)
97 \( 1 + (-5.88 + 5.88i)T - 97iT^{2} \)
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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

−11.54368933332736140165169666805, −9.849950322999013581071451053646, −9.220660818997293091339694201451, −8.459865251924565625578944630774, −6.80200783302131051393789034844, −5.82767020198965257977144142471, −5.06052256197369511611919327968, −3.75764132165778299803520356549, −2.37450833715834972879175534176, −0.834896008257784391687746633526, 2.93556765789158534541637526246, 3.91262064377299349460322485474, 4.73002593040089147369658578968, 6.40755694167855506090698331406, 6.79367103775611837113500624393, 7.58498707659901367272604724737, 9.092660234447984109781090933188, 10.00112324600442854408439282110, 10.77348565808689568364233491861, 11.96044227461390303898437033272

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