Properties

Degree $2$
Conductor $384$
Sign $-0.0537 - 0.998i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (−6.49 − 6.49i)5-s − 3.94·7-s + 2.99i·9-s + (4.31 − 4.31i)11-s + (−4.06 + 4.06i)13-s + 15.9i·15-s − 14.5·17-s + (4.94 + 4.94i)19-s + (4.82 + 4.82i)21-s + 43.6·23-s + 59.3i·25-s + (3.67 − 3.67i)27-s + (−25.0 + 25.0i)29-s + 32.5i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−1.29 − 1.29i)5-s − 0.563·7-s + 0.333i·9-s + (0.391 − 0.391i)11-s + (−0.312 + 0.312i)13-s + 1.06i·15-s − 0.856·17-s + (0.260 + 0.260i)19-s + (0.229 + 0.229i)21-s + 1.89·23-s + 2.37i·25-s + (0.136 − 0.136i)27-s + (−0.865 + 0.865i)29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0537 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0537 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.0537 - 0.998i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.0537 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.138427 + 0.146074i\)
\(L(\frac12)\) \(\approx\) \(0.138427 + 0.146074i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (6.49 + 6.49i)T + 25iT^{2} \)
7 \( 1 + 3.94T + 49T^{2} \)
11 \( 1 + (-4.31 + 4.31i)T - 121iT^{2} \)
13 \( 1 + (4.06 - 4.06i)T - 169iT^{2} \)
17 \( 1 + 14.5T + 289T^{2} \)
19 \( 1 + (-4.94 - 4.94i)T + 361iT^{2} \)
23 \( 1 - 43.6T + 529T^{2} \)
29 \( 1 + (25.0 - 25.0i)T - 841iT^{2} \)
31 \( 1 - 32.5iT - 961T^{2} \)
37 \( 1 + (4.14 + 4.14i)T + 1.36e3iT^{2} \)
41 \( 1 + 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (16.1 - 16.1i)T - 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (-31.5 - 31.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (49.7 - 49.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (44.4 - 44.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (1.64 + 1.64i)T + 4.48e3iT^{2} \)
71 \( 1 + 24.1T + 5.04e3T^{2} \)
73 \( 1 + 10.7iT - 5.32e3T^{2} \)
79 \( 1 + 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (-42.0 - 42.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−11.51096906822047255264876605025, −10.74022336375253412035685992928, −9.095890931735781864123425753837, −8.796287468717086480172513011781, −7.52536469233940950934302310542, −6.81375959351561525044243216229, −5.39399138171001066072876737462, −4.51664939785672203658917429112, −3.35425440506110556324159381341, −1.22054914330794762099385280590, 0.10589727605296832059992027055, 2.79934667803572398407410387399, 3.74286843356713870715205452707, 4.77879331189541433338982811476, 6.35475436747938331484649964719, 7.01223073764013465968319761538, 7.88170285914623857990836849271, 9.229294458101906697505640457179, 10.09669294568463470720723223018, 11.24309169825037293837162296554

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