Properties

Label 2-384-128.29-c1-0-24
Degree $2$
Conductor $384$
Sign $0.997 - 0.0758i$
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  + (1.41 − 0.0847i)2-s + (−0.0980 + 0.995i)3-s + (1.98 − 0.239i)4-s + (1.11 − 0.596i)5-s + (−0.0540 + 1.41i)6-s + (−0.322 − 1.62i)7-s + (2.78 − 0.505i)8-s + (−0.980 − 0.195i)9-s + (1.52 − 0.936i)10-s + (2.19 − 2.67i)11-s + (0.0433 + 1.99i)12-s + (−1.37 + 2.57i)13-s + (−0.592 − 2.26i)14-s + (0.484 + 1.16i)15-s + (3.88 − 0.949i)16-s + (−1.55 + 3.74i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0598i)2-s + (−0.0565 + 0.574i)3-s + (0.992 − 0.119i)4-s + (0.499 − 0.266i)5-s + (−0.0220 + 0.576i)6-s + (−0.121 − 0.613i)7-s + (0.983 − 0.178i)8-s + (−0.326 − 0.0650i)9-s + (0.482 − 0.296i)10-s + (0.661 − 0.806i)11-s + (0.0125 + 0.577i)12-s + (−0.381 + 0.714i)13-s + (−0.158 − 0.604i)14-s + (0.125 + 0.301i)15-s + (0.971 − 0.237i)16-s + (−0.376 + 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.997 - 0.0758i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58757 + 0.0982361i\)
\(L(\frac12)\) \(\approx\) \(2.58757 + 0.0982361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0847i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (-1.11 + 0.596i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.322 + 1.62i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-2.19 + 2.67i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (1.37 - 2.57i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (1.55 - 3.74i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.0648 - 0.213i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (3.78 - 2.52i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (3.94 - 3.24i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (-5.85 + 5.85i)T - 31iT^{2} \)
37 \( 1 + (11.2 + 3.40i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (-2.04 - 3.06i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.519 + 5.27i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (2.35 + 0.974i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (2.49 + 2.04i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (-3.50 - 6.55i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (-2.87 - 0.283i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (9.85 + 0.970i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (5.76 - 1.14i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.76 + 8.86i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (3.18 - 1.32i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (2.47 - 0.750i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-5.57 - 3.72i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (8.49 - 8.49i)T - 97iT^{2} \)
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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

−11.43679448730188647385029214072, −10.57584966152990381716747602373, −9.743056389004434806560645684879, −8.686901040023102762154847034086, −7.33603111737810674525604743440, −6.27225586571178520098173708176, −5.49246825988848352204448330210, −4.24995672384369274134958466038, −3.54576482097758448804452104262, −1.81901476901056059345186064336, 1.96660485528115156535369927580, 2.94276908070221634995993886811, 4.51522620636417741375779613282, 5.58274782422388526259327990279, 6.47531419489941776394009046833, 7.21296243764780598291332684668, 8.344102513576201715524066746112, 9.676582728910167257861157456529, 10.54351230646801447692054213829, 11.83631349876101055326470116823

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