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Properties

Label 2-384-16.11-c6-0-17
Degree $2$
Conductor $384$
Sign $0.876 - 0.480i$
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 + (−45.7 − 45.7i)5-s − 565.·7-s + 242. i·9-s + (−1.29e3 + 1.29e3i)11-s + (2.88e3 − 2.88e3i)13-s − 1.00e3i·15-s − 662.·17-s + (−1.94e3 − 1.94e3i)19-s + (−6.23e3 − 6.23e3i)21-s − 1.58e4·23-s − 1.14e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (−1.15e4 + 1.15e4i)29-s − 3.90e4i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.366 − 0.366i)5-s − 1.64·7-s + 0.333i·9-s + (−0.970 + 0.970i)11-s + (1.31 − 1.31i)13-s − 0.299i·15-s − 0.134·17-s + (−0.282 − 0.282i)19-s + (−0.673 − 0.673i)21-s − 1.30·23-s − 0.731i·25-s + (−0.136 + 0.136i)27-s + (−0.473 + 0.473i)29-s − 1.31i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.876 - 0.480i$
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: \((2,\ 384,\ (\ :3),\ 0.876 - 0.480i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.204593141\)
\(L(\frac12)\) \(\approx\) \(1.204593141\)
\(L(4)\) 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 \( 1 + (45.7 + 45.7i)T + 1.56e4iT^{2} \)
7 \( 1 + 565.T + 1.17e5T^{2} \)
11 \( 1 + (1.29e3 - 1.29e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (-2.88e3 + 2.88e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 662.T + 2.41e7T^{2} \)
19 \( 1 + (1.94e3 + 1.94e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.58e4T + 1.48e8T^{2} \)
29 \( 1 + (1.15e4 - 1.15e4i)T - 5.94e8iT^{2} \)
31 \( 1 + 3.90e4iT - 8.87e8T^{2} \)
37 \( 1 + (-5.13e4 - 5.13e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 6.65e4iT - 4.75e9T^{2} \)
43 \( 1 + (6.89e4 - 6.89e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 1.07e5iT - 1.07e10T^{2} \)
53 \( 1 + (-1.08e5 - 1.08e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (-6.78e4 + 6.78e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (-9.92e4 + 9.92e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (-1.05e5 - 1.05e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.66e5T + 1.28e11T^{2} \)
73 \( 1 - 2.67e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.57e5iT - 2.43e11T^{2} \)
83 \( 1 + (-2.36e5 - 2.36e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 4.19e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.93e5T + 8.32e11T^{2} \)
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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

−10.01197707256117840174148031718, −9.826084502449494571463927396481, −8.463692173439445138639260941028, −7.86279358297101983850315235153, −6.55350359133772204598332986360, −5.63582115618580929590574109739, −4.31914984763532306069291049934, −3.39790768756603460122632142977, −2.43009271101412863051005181942, −0.56731918462850900836967757916, 0.45390272347836292795857406395, 2.08011642571566792944530346595, 3.32736282224206595845607873115, 3.85606374110657280326491304641, 5.81948512547120999086384630130, 6.45789237148250983312404730352, 7.38397725522368497407146901070, 8.496944367164427011014246492012, 9.204518524260904182776571453107, 10.27090716096551474477213299133

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