Properties

Label 2-384-128.107-c2-0-24
Degree $2$
Conductor $384$
Sign $0.883 - 0.468i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.183 + 1.99i)2-s + (1.09 − 1.33i)3-s + (−3.93 − 0.729i)4-s + (−1.55 − 5.11i)5-s + (2.46 + 2.43i)6-s + (−2.38 + 11.9i)7-s + (2.17 − 7.69i)8-s + (−0.585 − 2.94i)9-s + (10.4 − 2.15i)10-s + (13.5 + 1.33i)11-s + (−5.29 + 4.46i)12-s + (−1.09 − 0.332i)13-s + (−23.4 − 6.93i)14-s + (−8.55 − 3.54i)15-s + (14.9 + 5.73i)16-s + (−10.0 − 24.1i)17-s + ⋯
L(s)  = 1  + (−0.0915 + 0.995i)2-s + (0.366 − 0.446i)3-s + (−0.983 − 0.182i)4-s + (−0.310 − 1.02i)5-s + (0.410 + 0.405i)6-s + (−0.340 + 1.71i)7-s + (0.271 − 0.962i)8-s + (−0.0650 − 0.326i)9-s + (1.04 − 0.215i)10-s + (1.23 + 0.121i)11-s + (−0.441 + 0.372i)12-s + (−0.0843 − 0.0255i)13-s + (−1.67 − 0.495i)14-s + (−0.570 − 0.236i)15-s + (0.933 + 0.358i)16-s + (−0.589 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55003 + 0.385440i\)
\(L(\frac12)\) \(\approx\) \(1.55003 + 0.385440i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.183 - 1.99i)T \)
3 \( 1 + (-1.09 + 1.33i)T \)
good5 \( 1 + (1.55 + 5.11i)T + (-20.7 + 13.8i)T^{2} \)
7 \( 1 + (2.38 - 11.9i)T + (-45.2 - 18.7i)T^{2} \)
11 \( 1 + (-13.5 - 1.33i)T + (118. + 23.6i)T^{2} \)
13 \( 1 + (1.09 + 0.332i)T + (140. + 93.8i)T^{2} \)
17 \( 1 + (10.0 + 24.1i)T + (-204. + 204. i)T^{2} \)
19 \( 1 + (-17.0 + 9.09i)T + (200. - 300. i)T^{2} \)
23 \( 1 + (-36.7 - 24.5i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (-24.2 + 2.39i)T + (824. - 164. i)T^{2} \)
31 \( 1 + (-13.8 + 13.8i)T - 961iT^{2} \)
37 \( 1 + (-31.5 - 16.8i)T + (760. + 1.13e3i)T^{2} \)
41 \( 1 + (-10.3 - 6.89i)T + (643. + 1.55e3i)T^{2} \)
43 \( 1 + (7.80 + 9.51i)T + (-360. + 1.81e3i)T^{2} \)
47 \( 1 + (10.6 + 25.6i)T + (-1.56e3 + 1.56e3i)T^{2} \)
53 \( 1 + (-79.8 - 7.86i)T + (2.75e3 + 548. i)T^{2} \)
59 \( 1 + (-9.87 - 32.5i)T + (-2.89e3 + 1.93e3i)T^{2} \)
61 \( 1 + (36.0 - 43.8i)T + (-725. - 3.64e3i)T^{2} \)
67 \( 1 + (73.9 + 60.6i)T + (875. + 4.40e3i)T^{2} \)
71 \( 1 + (91.1 + 18.1i)T + (4.65e3 + 1.92e3i)T^{2} \)
73 \( 1 + (-22.2 + 4.43i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (-20.6 + 49.8i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (-45.1 - 84.3i)T + (-3.82e3 + 5.72e3i)T^{2} \)
89 \( 1 + (-14.2 - 21.2i)T + (-3.03e3 + 7.31e3i)T^{2} \)
97 \( 1 + (-19.0 - 19.0i)T + 9.40e3iT^{2} \)
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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

−11.73601636796105227034446961264, −9.491794392544087437733359961002, −9.124552004468497292038127304312, −8.618659764179906525800162197476, −7.41611905083577421362413657764, −6.51981667211821654562102938898, −5.42762155354635215937251493678, −4.61593636172397288373022320646, −2.95051424011840499017474968069, −0.973793889637095610995311989766, 1.11302280850969806285234838380, 2.99526983905883382878941531083, 3.79389051862900921467519517136, 4.49877965268979014482173134416, 6.47098271223523999914372314673, 7.35292565383273624493581555893, 8.498992187796055860727825152215, 9.495759571882873863938665904889, 10.53838547754892286030760403017, 10.67539571230031342970399951684

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