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Properties

Label 2-384-48.5-c2-0-2
Degree $2$
Conductor $384$
Sign $-0.606 + 0.795i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.14 + 2.77i)3-s + (−4.80 + 4.80i)5-s + 7.36i·7-s + (−6.35 − 6.37i)9-s + (0.514 − 0.514i)11-s + (−7.12 + 7.12i)13-s + (−7.79 − 18.8i)15-s + 11.1i·17-s + (21.1 − 21.1i)19-s + (−20.4 − 8.46i)21-s − 7.80·23-s − 21.1i·25-s + (24.9 − 10.2i)27-s + (−34.6 − 34.6i)29-s + 24.8·31-s + ⋯
L(s)  = 1  + (−0.383 + 0.923i)3-s + (−0.960 + 0.960i)5-s + 1.05i·7-s + (−0.706 − 0.707i)9-s + (0.0467 − 0.0467i)11-s + (−0.548 + 0.548i)13-s + (−0.519 − 1.25i)15-s + 0.653i·17-s + (1.11 − 1.11i)19-s + (−0.971 − 0.402i)21-s − 0.339·23-s − 0.846i·25-s + (0.924 − 0.381i)27-s + (−1.19 − 1.19i)29-s + 0.802·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.606 + 0.795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.192417 - 0.388551i\)
\(L(\frac12)\) \(\approx\) \(0.192417 - 0.388551i\)
\(L(2)\) 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 \( 1 + (1.14 - 2.77i)T \)
good5 \( 1 + (4.80 - 4.80i)T - 25iT^{2} \)
7 \( 1 - 7.36iT - 49T^{2} \)
11 \( 1 + (-0.514 + 0.514i)T - 121iT^{2} \)
13 \( 1 + (7.12 - 7.12i)T - 169iT^{2} \)
17 \( 1 - 11.1iT - 289T^{2} \)
19 \( 1 + (-21.1 + 21.1i)T - 361iT^{2} \)
23 \( 1 + 7.80T + 529T^{2} \)
29 \( 1 + (34.6 + 34.6i)T + 841iT^{2} \)
31 \( 1 - 24.8T + 961T^{2} \)
37 \( 1 + (-18.2 - 18.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 64.2T + 1.68e3T^{2} \)
43 \( 1 + (7.24 + 7.24i)T + 1.84e3iT^{2} \)
47 \( 1 + 23.0iT - 2.20e3T^{2} \)
53 \( 1 + (31.9 - 31.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (17.6 - 17.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-12.3 + 12.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (41.1 - 41.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 25.6T + 5.04e3T^{2} \)
73 \( 1 + 56.1iT - 5.32e3T^{2} \)
79 \( 1 + 35.7T + 6.24e3T^{2} \)
83 \( 1 + (-94.9 - 94.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 44.8T + 7.92e3T^{2} \)
97 \( 1 + 82.3T + 9.40e3T^{2} \)
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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

−11.77244392403630209633541062196, −10.93791124608748987030161918411, −9.910484834753022741105186973135, −9.126706913696105040936240852568, −8.059644957608627540073092247851, −6.93077847901291770380509074838, −5.91301962154416351062243845474, −4.80159971253708434736054649240, −3.68182050467284809794760207548, −2.65130800829119579363150659520, 0.21447102864772696759487961378, 1.35416425487326844351342360959, 3.36471876734153677540060582892, 4.65366247408984671588203252683, 5.57492423084455305881190087306, 7.03315140034452471310712983878, 7.66755434339717982889984047782, 8.312930943516032908929379743423, 9.634970151213935427057861991240, 10.71541716203870702464727345101

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