Properties

Label 2-384-128.13-c1-0-19
Degree $2$
Conductor $384$
Sign $0.599 - 0.800i$
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  + (1.31 + 0.523i)2-s + (−0.290 − 0.956i)3-s + (1.45 + 1.37i)4-s + (0.344 + 3.50i)5-s + (0.119 − 1.40i)6-s + (1.21 − 1.81i)7-s + (1.18 + 2.56i)8-s + (−0.831 + 0.555i)9-s + (−1.38 + 4.78i)10-s + (0.838 − 0.448i)11-s + (0.895 − 1.78i)12-s + (−4.53 − 0.446i)13-s + (2.54 − 1.75i)14-s + (3.25 − 1.34i)15-s + (0.212 + 3.99i)16-s + (5.82 + 2.41i)17-s + ⋯
L(s)  = 1  + (0.928 + 0.370i)2-s + (−0.167 − 0.552i)3-s + (0.725 + 0.688i)4-s + (0.154 + 1.56i)5-s + (0.0489 − 0.575i)6-s + (0.458 − 0.686i)7-s + (0.419 + 0.907i)8-s + (−0.277 + 0.185i)9-s + (−0.436 + 1.51i)10-s + (0.252 − 0.135i)11-s + (0.258 − 0.516i)12-s + (−1.25 − 0.123i)13-s + (0.680 − 0.467i)14-s + (0.839 − 0.347i)15-s + (0.0532 + 0.998i)16-s + (1.41 + 0.584i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.599 - 0.800i$
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: \((2,\ 384,\ (\ :1/2),\ 0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06472 + 1.03284i\)
\(L(\frac12)\) \(\approx\) \(2.06472 + 1.03284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 0.523i)T \)
3 \( 1 + (0.290 + 0.956i)T \)
good5 \( 1 + (-0.344 - 3.50i)T + (-4.90 + 0.975i)T^{2} \)
7 \( 1 + (-1.21 + 1.81i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-0.838 + 0.448i)T + (6.11 - 9.14i)T^{2} \)
13 \( 1 + (4.53 + 0.446i)T + (12.7 + 2.53i)T^{2} \)
17 \( 1 + (-5.82 - 2.41i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (2.76 - 2.26i)T + (3.70 - 18.6i)T^{2} \)
23 \( 1 + (-1.01 + 5.12i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-2.86 + 5.35i)T + (-16.1 - 24.1i)T^{2} \)
31 \( 1 + (-5.07 + 5.07i)T - 31iT^{2} \)
37 \( 1 + (-0.734 + 0.895i)T + (-7.21 - 36.2i)T^{2} \)
41 \( 1 + (7.43 + 1.47i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.662 + 2.18i)T + (-35.7 - 23.8i)T^{2} \)
47 \( 1 + (-0.0750 + 0.181i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (6.16 + 11.5i)T + (-29.4 + 44.0i)T^{2} \)
59 \( 1 + (4.87 - 0.479i)T + (57.8 - 11.5i)T^{2} \)
61 \( 1 + (-0.0109 + 0.00331i)T + (50.7 - 33.8i)T^{2} \)
67 \( 1 + (10.8 - 3.29i)T + (55.7 - 37.2i)T^{2} \)
71 \( 1 + (6.04 + 4.04i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (5.66 + 8.47i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (2.80 + 6.76i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-10.2 - 12.4i)T + (-16.1 + 81.4i)T^{2} \)
89 \( 1 + (-2.36 - 11.9i)T + (-82.2 + 34.0i)T^{2} \)
97 \( 1 + (-5.26 + 5.26i)T - 97iT^{2} \)
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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

−11.70326722709574682543050357884, −10.62641757423709497646272839686, −10.14194746765071988890193122789, −8.072729438059348103821455013258, −7.52149955977487317647338132779, −6.62564088696570369877582971212, −5.96161152915410052350041482061, −4.56708946742858715753166596919, −3.30585777553988378071954120871, −2.21172083412659898189714503813, 1.43027606261260462571317415411, 3.04157914526292111075891391562, 4.72722567558500481709087637840, 4.91840421219229816266037564466, 5.86809368050609305530476625680, 7.36798389380314871708291876116, 8.685892124382849336377583534980, 9.507193239634559159094698817221, 10.27686644388590908529177963957, 11.65292546101161109987822006063

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