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Properties

Label 2-384-16.11-c6-0-46
Degree $2$
Conductor $384$
Sign $0.382 + 0.924i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.0 + 11.0i)3-s + (−29.4 − 29.4i)5-s + 461.·7-s + 242. i·9-s + (1.18e3 − 1.18e3i)11-s + (2.43e3 − 2.43e3i)13-s − 650. i·15-s + 4.81e3·17-s + (−8.31e3 − 8.31e3i)19-s + (5.08e3 + 5.08e3i)21-s − 8.17e3·23-s − 1.38e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (−1.67e4 + 1.67e4i)29-s + 7.90e3i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.235 − 0.235i)5-s + 1.34·7-s + 0.333i·9-s + (0.891 − 0.891i)11-s + (1.10 − 1.10i)13-s − 0.192i·15-s + 0.980·17-s + (−1.21 − 1.21i)19-s + (0.548 + 0.548i)21-s − 0.672·23-s − 0.888i·25-s + (−0.136 + 0.136i)27-s + (−0.688 + 0.688i)29-s + 0.265i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.382 + 0.924i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ 0.382 + 0.924i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.899758512\)
\(L(\frac12)\) \(\approx\) \(2.899758512\)
\(L(4)\) 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 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (29.4 + 29.4i)T + 1.56e4iT^{2} \)
7 \( 1 - 461.T + 1.17e5T^{2} \)
11 \( 1 + (-1.18e3 + 1.18e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (-2.43e3 + 2.43e3i)T - 4.82e6iT^{2} \)
17 \( 1 - 4.81e3T + 2.41e7T^{2} \)
19 \( 1 + (8.31e3 + 8.31e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 8.17e3T + 1.48e8T^{2} \)
29 \( 1 + (1.67e4 - 1.67e4i)T - 5.94e8iT^{2} \)
31 \( 1 - 7.90e3iT - 8.87e8T^{2} \)
37 \( 1 + (2.21e4 + 2.21e4i)T + 2.56e9iT^{2} \)
41 \( 1 + 7.14e4iT - 4.75e9T^{2} \)
43 \( 1 + (7.07e4 - 7.07e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 3.15e3iT - 1.07e10T^{2} \)
53 \( 1 + (-2.04e3 - 2.04e3i)T + 2.21e10iT^{2} \)
59 \( 1 + (-2.55e5 + 2.55e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (2.86e5 - 2.86e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (2.92e5 + 2.92e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 1.28e5T + 1.28e11T^{2} \)
73 \( 1 + 7.98e4iT - 1.51e11T^{2} \)
79 \( 1 - 1.38e5iT - 2.43e11T^{2} \)
83 \( 1 + (-1.36e5 - 1.36e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 - 4.69e4T + 8.32e11T^{2} \)
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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

−10.36050571333057785378587382217, −8.867610274234072150209812328752, −8.495878099075011741254808034869, −7.67449252907717238764188317099, −6.22267726359568391824800296315, −5.20562391958462510072556166027, −4.16378982157216243850296239269, −3.23661503913479830817722808895, −1.72275682245290118089401840897, −0.61107211091438171521136649715, 1.48206774862643283861997937061, 1.83587049644656539801841721468, 3.69282677428165425011183308936, 4.37863225276483283024808917735, 5.84812347740196425419569708603, 6.84794861723056479936240030264, 7.83293701395121234166839472772, 8.477921683466711574247554417779, 9.480230859122159143519010470455, 10.56598901018772304575351648292

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