Properties

Degree $2$
Conductor $384$
Sign $0.582 - 0.812i$
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 + (3.40 + 3.40i)5-s + 12.1·7-s + 2.99i·9-s + (−9.81 + 9.81i)11-s + (7.76 − 7.76i)13-s + 8.34i·15-s + 9.73·17-s + (−11.2 − 11.2i)19-s + (14.8 + 14.8i)21-s − 20.2·23-s − 1.80i·25-s + (−3.67 + 3.67i)27-s + (16.4 − 16.4i)29-s + 26.3i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.681 + 0.681i)5-s + 1.73·7-s + 0.333i·9-s + (−0.891 + 0.891i)11-s + (0.597 − 0.597i)13-s + 0.556i·15-s + 0.572·17-s + (−0.593 − 0.593i)19-s + (0.707 + 0.707i)21-s − 0.881·23-s − 0.0720i·25-s + (−0.136 + 0.136i)27-s + (0.565 − 0.565i)29-s + 0.850i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.582 - 0.812i$
Motivic weight: \(2\)
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.582 - 0.812i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.22602 + 1.14277i\)
\(L(\frac12)\) \(\approx\) \(2.22602 + 1.14277i\)
\(L(2)\) 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 \( 1 + (-3.40 - 3.40i)T + 25iT^{2} \)
7 \( 1 - 12.1T + 49T^{2} \)
11 \( 1 + (9.81 - 9.81i)T - 121iT^{2} \)
13 \( 1 + (-7.76 + 7.76i)T - 169iT^{2} \)
17 \( 1 - 9.73T + 289T^{2} \)
19 \( 1 + (11.2 + 11.2i)T + 361iT^{2} \)
23 \( 1 + 20.2T + 529T^{2} \)
29 \( 1 + (-16.4 + 16.4i)T - 841iT^{2} \)
31 \( 1 - 26.3iT - 961T^{2} \)
37 \( 1 + (-23.7 - 23.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 24.7iT - 1.68e3T^{2} \)
43 \( 1 + (29.8 - 29.8i)T - 1.84e3iT^{2} \)
47 \( 1 - 31.3iT - 2.20e3T^{2} \)
53 \( 1 + (36.8 + 36.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (-14.1 + 14.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (-42.5 + 42.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (48.7 + 48.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 7.73T + 5.04e3T^{2} \)
73 \( 1 + 85.4iT - 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 + (-62.1 - 62.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−10.95633438737325008737812226585, −10.43545962317090483971028555336, −9.618224243415068771284644731362, −8.180871633338468650502485785880, −7.917573639762129751528553246720, −6.47914443782663112661237509802, −5.23821047479126500002145739450, −4.46220271292802524183529127639, −2.82393772953858171542560918417, −1.76847110282858114141954822196, 1.24020761589860095991887483509, 2.22748220124212096087195567804, 4.00275867541430276413253591705, 5.24474673519995267023307100297, 5.96037972849180683598601163933, 7.51370913100257810363480341709, 8.348912856870365687323489210919, 8.780163740197374570483461507247, 10.10361348027720444401679530453, 11.04971730473984133507698192586

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