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Properties

Label 2-384-48.35-c3-0-29
Degree $2$
Conductor $384$
Sign $0.232 + 0.972i$
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  + (2.71 + 4.43i)3-s + (−3.17 − 3.17i)5-s − 32.3·7-s + (−12.2 + 24.0i)9-s + (16.0 − 16.0i)11-s + (18.2 + 18.2i)13-s + (5.45 − 22.6i)15-s + 38.5i·17-s + (56.2 − 56.2i)19-s + (−87.7 − 143. i)21-s − 197. i·23-s − 104. i·25-s + (−139. + 10.7i)27-s + (57.3 − 57.3i)29-s − 148. i·31-s + ⋯
L(s)  = 1  + (0.522 + 0.852i)3-s + (−0.284 − 0.284i)5-s − 1.74·7-s + (−0.454 + 0.890i)9-s + (0.441 − 0.441i)11-s + (0.390 + 0.390i)13-s + (0.0939 − 0.390i)15-s + 0.550i·17-s + (0.679 − 0.679i)19-s + (−0.911 − 1.48i)21-s − 1.79i·23-s − 0.838i·25-s + (−0.997 + 0.0768i)27-s + (0.366 − 0.366i)29-s − 0.859i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.232 + 0.972i$
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: \((2,\ 384,\ (\ :3/2),\ 0.232 + 0.972i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9586920928\)
\(L(\frac12)\) \(\approx\) \(0.9586920928\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.71 - 4.43i)T \)
good5 \( 1 + (3.17 + 3.17i)T + 125iT^{2} \)
7 \( 1 + 32.3T + 343T^{2} \)
11 \( 1 + (-16.0 + 16.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-18.2 - 18.2i)T + 2.19e3iT^{2} \)
17 \( 1 - 38.5iT - 4.91e3T^{2} \)
19 \( 1 + (-56.2 + 56.2i)T - 6.85e3iT^{2} \)
23 \( 1 + 197. iT - 1.21e4T^{2} \)
29 \( 1 + (-57.3 + 57.3i)T - 2.43e4iT^{2} \)
31 \( 1 + 148. iT - 2.97e4T^{2} \)
37 \( 1 + (-72.5 + 72.5i)T - 5.06e4iT^{2} \)
41 \( 1 + 73.1T + 6.89e4T^{2} \)
43 \( 1 + (226. + 226. i)T + 7.95e4iT^{2} \)
47 \( 1 + 412.T + 1.03e5T^{2} \)
53 \( 1 + (-94.8 - 94.8i)T + 1.48e5iT^{2} \)
59 \( 1 + (344. - 344. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-153. - 153. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-603. + 603. i)T - 3.00e5iT^{2} \)
71 \( 1 - 711. iT - 3.57e5T^{2} \)
73 \( 1 + 687. iT - 3.89e5T^{2} \)
79 \( 1 - 162. iT - 4.93e5T^{2} \)
83 \( 1 + (748. + 748. i)T + 5.71e5iT^{2} \)
89 \( 1 + 927.T + 7.04e5T^{2} \)
97 \( 1 + 208.T + 9.12e5T^{2} \)
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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

−10.47652356656583113954890911201, −9.765370915637344859434934807954, −8.960734451854477763117024798669, −8.264834455128677107964496021833, −6.78778509345134405848251381054, −5.97262373592351219885008779067, −4.48924975910833731229647086361, −3.62451141214677442985493595888, −2.64004833826752819363106681753, −0.31701606084233806960502673601, 1.34259349005496329661420323485, 3.11406587837066427994342335396, 3.52329353703097748901287666885, 5.56099646993005302740331051663, 6.64622217056793856259106276722, 7.17972420308001083262101694293, 8.229504702409470104493258211480, 9.479175853241382831990123669818, 9.792463131502829050754088099346, 11.32693281920654578887021585503

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