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Properties

Label 2-384-128.29-c1-0-21
Degree $2$
Conductor $384$
Sign $0.0868 + 0.996i$
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  + (−1.15 − 0.812i)2-s + (−0.0980 + 0.995i)3-s + (0.680 + 1.88i)4-s + (1.63 − 0.873i)5-s + (0.921 − 1.07i)6-s + (−0.848 − 4.26i)7-s + (0.738 − 2.73i)8-s + (−0.980 − 0.195i)9-s + (−2.59 − 0.315i)10-s + (0.548 − 0.668i)11-s + (−1.93 + 0.493i)12-s + (−2.49 + 4.67i)13-s + (−2.48 + 5.62i)14-s + (0.708 + 1.71i)15-s + (−3.07 + 2.56i)16-s + (2.00 − 4.84i)17-s + ⋯
L(s)  = 1  + (−0.818 − 0.574i)2-s + (−0.0565 + 0.574i)3-s + (0.340 + 0.940i)4-s + (0.730 − 0.390i)5-s + (0.376 − 0.437i)6-s + (−0.320 − 1.61i)7-s + (0.261 − 0.965i)8-s + (−0.326 − 0.0650i)9-s + (−0.822 − 0.0998i)10-s + (0.165 − 0.201i)11-s + (−0.559 + 0.142i)12-s + (−0.692 + 1.29i)13-s + (−0.663 + 1.50i)14-s + (0.182 + 0.441i)15-s + (−0.768 + 0.640i)16-s + (0.486 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0868 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0868 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.0868 + 0.996i$
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: \((2,\ 384,\ (\ :1/2),\ 0.0868 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661161 - 0.606053i\)
\(L(\frac12)\) \(\approx\) \(0.661161 - 0.606053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.812i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (-1.63 + 0.873i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.848 + 4.26i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.548 + 0.668i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (2.49 - 4.67i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (-2.00 + 4.84i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (1.81 + 5.99i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (-7.85 + 5.25i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (1.84 - 1.51i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (-1.28 + 1.28i)T - 31iT^{2} \)
37 \( 1 + (-0.952 - 0.288i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (4.40 + 6.59i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.205 + 2.08i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (-7.86 - 3.25i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-5.54 - 4.55i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (0.529 + 0.989i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (-8.55 - 0.842i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (-12.0 - 1.19i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (6.55 - 1.30i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.24 + 6.24i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (11.6 - 4.81i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (14.0 - 4.26i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-4.21 - 2.81i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-1.39 + 1.39i)T - 97iT^{2} \)
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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

−10.92091649481340318476132794958, −10.15101983581092132901927387392, −9.349292638118531914827181636416, −8.878776848338241344486746229805, −7.21810782879968454990240392402, −6.83855551062840937007853847748, −4.93227594021444531552482384787, −4.00572547551523201728261125738, −2.60898073325325038070326905854, −0.807509750557929729841012941534, 1.74977217933597588064299014129, 2.86777968197858700599116594561, 5.55280149364203060150590563826, 5.74446298259607145703861042790, 6.81412920121372149046904402898, 7.960582173952663501616939229575, 8.650491624216708858045596410803, 9.747241836982517897761301153836, 10.26930544638963295897532403750, 11.50293698827239873188418462383

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