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Properties

Label 2-384-16.13-c3-0-0
Degree $2$
Conductor $384$
Sign $-0.909 - 0.415i$
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  + (−2.12 − 2.12i)3-s + (7.29 − 7.29i)5-s + 22.1i·7-s + 8.99i·9-s + (−8.24 + 8.24i)11-s + (−51.9 − 51.9i)13-s − 30.9·15-s − 58.7·17-s + (54.5 + 54.5i)19-s + (47.0 − 47.0i)21-s − 117. i·23-s + 18.6i·25-s + (19.0 − 19.0i)27-s + (−175. − 175. i)29-s − 6.58·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.652 − 0.652i)5-s + 1.19i·7-s + 0.333i·9-s + (−0.225 + 0.225i)11-s + (−1.10 − 1.10i)13-s − 0.532·15-s − 0.837·17-s + (0.658 + 0.658i)19-s + (0.488 − 0.488i)21-s − 1.06i·23-s + 0.149i·25-s + (0.136 − 0.136i)27-s + (−1.12 − 1.12i)29-s − 0.0381·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.909 - 0.415i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.909 - 0.415i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.05623827396\)
\(L(\frac12)\) \(\approx\) \(0.05623827396\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (-7.29 + 7.29i)T - 125iT^{2} \)
7 \( 1 - 22.1iT - 343T^{2} \)
11 \( 1 + (8.24 - 8.24i)T - 1.33e3iT^{2} \)
13 \( 1 + (51.9 + 51.9i)T + 2.19e3iT^{2} \)
17 \( 1 + 58.7T + 4.91e3T^{2} \)
19 \( 1 + (-54.5 - 54.5i)T + 6.85e3iT^{2} \)
23 \( 1 + 117. iT - 1.21e4T^{2} \)
29 \( 1 + (175. + 175. i)T + 2.43e4iT^{2} \)
31 \( 1 + 6.58T + 2.97e4T^{2} \)
37 \( 1 + (265. - 265. i)T - 5.06e4iT^{2} \)
41 \( 1 - 98.7iT - 6.89e4T^{2} \)
43 \( 1 + (347. - 347. i)T - 7.95e4iT^{2} \)
47 \( 1 + 141.T + 1.03e5T^{2} \)
53 \( 1 + (-210. + 210. i)T - 1.48e5iT^{2} \)
59 \( 1 + (427. - 427. i)T - 2.05e5iT^{2} \)
61 \( 1 + (178. + 178. i)T + 2.26e5iT^{2} \)
67 \( 1 + (480. + 480. i)T + 3.00e5iT^{2} \)
71 \( 1 - 884. iT - 3.57e5T^{2} \)
73 \( 1 + 794. iT - 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 + (-167. - 167. i)T + 5.71e5iT^{2} \)
89 \( 1 - 664. iT - 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−11.53410635375194526066710914636, −10.25801198133671659770285620028, −9.530421409021312312873723336379, −8.538551320664961689916423500576, −7.65351050127864562863500610336, −6.35235564258472256826168679424, −5.47163708671251079021919340848, −4.84065672803695045896912195553, −2.80796150722676309304402943555, −1.73288588556421249429950529284, 0.01856632018706001315484589915, 1.90547158269814492408451372919, 3.42778785295215483450320784650, 4.57687120303847671705775143647, 5.59760631379387218300915553254, 7.00830235180128799990422757584, 7.17794825568082716828601050872, 8.978783745159269406113112870463, 9.747605002620621853794948264561, 10.57198678374818828197361318230

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