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Properties

Label 2-384-48.35-c3-0-9
Degree $2$
Conductor $384$
Sign $-0.995 - 0.0900i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.151 + 5.19i)3-s + (4.66 + 4.66i)5-s + 0.405·7-s + (−26.9 + 1.56i)9-s + (−5.82 + 5.82i)11-s + (35.2 + 35.2i)13-s + (−23.5 + 24.9i)15-s − 49.3i·17-s + (−108. + 108. i)19-s + (0.0612 + 2.10i)21-s + 130. i·23-s − 81.4i·25-s + (−12.2 − 139. i)27-s + (−172. + 172. i)29-s + 36.1i·31-s + ⋯
L(s)  = 1  + (0.0290 + 0.999i)3-s + (0.417 + 0.417i)5-s + 0.0219·7-s + (−0.998 + 0.0581i)9-s + (−0.159 + 0.159i)11-s + (0.751 + 0.751i)13-s + (−0.405 + 0.429i)15-s − 0.703i·17-s + (−1.30 + 1.30i)19-s + (0.000636 + 0.0219i)21-s + 1.18i·23-s − 0.651i·25-s + (−0.0870 − 0.996i)27-s + (−1.10 + 1.10i)29-s + 0.209i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0900i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.995 - 0.0900i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.995 - 0.0900i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.216828268\)
\(L(\frac12)\) \(\approx\) \(1.216828268\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.151 - 5.19i)T \)
good5 \( 1 + (-4.66 - 4.66i)T + 125iT^{2} \)
7 \( 1 - 0.405T + 343T^{2} \)
11 \( 1 + (5.82 - 5.82i)T - 1.33e3iT^{2} \)
13 \( 1 + (-35.2 - 35.2i)T + 2.19e3iT^{2} \)
17 \( 1 + 49.3iT - 4.91e3T^{2} \)
19 \( 1 + (108. - 108. i)T - 6.85e3iT^{2} \)
23 \( 1 - 130. iT - 1.21e4T^{2} \)
29 \( 1 + (172. - 172. i)T - 2.43e4iT^{2} \)
31 \( 1 - 36.1iT - 2.97e4T^{2} \)
37 \( 1 + (-257. + 257. i)T - 5.06e4iT^{2} \)
41 \( 1 - 5.87T + 6.89e4T^{2} \)
43 \( 1 + (170. + 170. i)T + 7.95e4iT^{2} \)
47 \( 1 - 181.T + 1.03e5T^{2} \)
53 \( 1 + (148. + 148. i)T + 1.48e5iT^{2} \)
59 \( 1 + (567. - 567. i)T - 2.05e5iT^{2} \)
61 \( 1 + (481. + 481. i)T + 2.26e5iT^{2} \)
67 \( 1 + (296. - 296. i)T - 3.00e5iT^{2} \)
71 \( 1 - 533. iT - 3.57e5T^{2} \)
73 \( 1 + 178. iT - 3.89e5T^{2} \)
79 \( 1 + 528. iT - 4.93e5T^{2} \)
83 \( 1 + (-713. - 713. i)T + 5.71e5iT^{2} \)
89 \( 1 + 204.T + 7.04e5T^{2} \)
97 \( 1 - 275.T + 9.12e5T^{2} \)
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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

−11.07963826014998979484855909991, −10.49863779891844674893546626673, −9.546628046509149243452793017301, −8.863473691505192766877069413824, −7.72528505004784577062155659734, −6.39824102041259634924794512357, −5.57642168478600072651737790317, −4.34040253872039246768288079261, −3.38855335844599498484302776564, −1.95504066206612208046950943826, 0.39120354262212633086832587662, 1.73895277824286378278267837661, 2.97022779615736640556015858736, 4.59358866826793727581665384462, 5.92012067315301823709075910001, 6.46465094879234281922945712894, 7.80282399423534200017529575215, 8.473505950618213224977702910532, 9.352456111978019751136179346485, 10.70851960597963371246317058849

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