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Properties

Label 2-384-48.11-c3-0-36
Degree $2$
Conductor $384$
Sign $0.910 + 0.412i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.81 + 1.96i)3-s + (6.30 − 6.30i)5-s + 24.6·7-s + (19.3 + 18.8i)9-s + (−40.4 − 40.4i)11-s + (47.3 − 47.3i)13-s + (42.6 − 17.9i)15-s − 41.7i·17-s + (−10.6 − 10.6i)19-s + (118. + 48.3i)21-s + 53.4i·23-s + 45.5i·25-s + (55.9 + 128. i)27-s + (−105. − 105. i)29-s − 3.14i·31-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)3-s + (0.563 − 0.563i)5-s + 1.33·7-s + (0.715 + 0.698i)9-s + (−1.10 − 1.10i)11-s + (1.00 − 1.00i)13-s + (0.734 − 0.309i)15-s − 0.595i·17-s + (−0.128 − 0.128i)19-s + (1.23 + 0.502i)21-s + 0.484i·23-s + 0.364i·25-s + (0.398 + 0.917i)27-s + (−0.674 − 0.674i)29-s − 0.0182i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.329524309\)
\(L(\frac12)\) \(\approx\) \(3.329524309\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.81 - 1.96i)T \)
good5 \( 1 + (-6.30 + 6.30i)T - 125iT^{2} \)
7 \( 1 - 24.6T + 343T^{2} \)
11 \( 1 + (40.4 + 40.4i)T + 1.33e3iT^{2} \)
13 \( 1 + (-47.3 + 47.3i)T - 2.19e3iT^{2} \)
17 \( 1 + 41.7iT - 4.91e3T^{2} \)
19 \( 1 + (10.6 + 10.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 53.4iT - 1.21e4T^{2} \)
29 \( 1 + (105. + 105. i)T + 2.43e4iT^{2} \)
31 \( 1 + 3.14iT - 2.97e4T^{2} \)
37 \( 1 + (42.1 + 42.1i)T + 5.06e4iT^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 + (221. - 221. i)T - 7.95e4iT^{2} \)
47 \( 1 - 381.T + 1.03e5T^{2} \)
53 \( 1 + (-294. + 294. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-445. - 445. i)T + 2.05e5iT^{2} \)
61 \( 1 + (21.8 - 21.8i)T - 2.26e5iT^{2} \)
67 \( 1 + (-572. - 572. i)T + 3.00e5iT^{2} \)
71 \( 1 - 612. iT - 3.57e5T^{2} \)
73 \( 1 + 331. iT - 3.89e5T^{2} \)
79 \( 1 - 427. iT - 4.93e5T^{2} \)
83 \( 1 + (245. - 245. i)T - 5.71e5iT^{2} \)
89 \( 1 - 188.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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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

−10.79981841202849126337675361356, −9.918639296022189122412032282721, −8.741285127049941538199551989302, −8.301861163349211806098936095559, −7.50191038672819767475115891813, −5.61800309698045831226691518782, −5.09845060733553475564414818028, −3.70315891202079453486192982731, −2.46731506941494977122905402553, −1.13335104129019027709170951860, 1.67152952106309520648428953463, 2.29387081513357488980567845732, 3.87504867677135257050144678382, 5.00467376533074268966319051006, 6.41700135084363315494855457771, 7.33416664405271283892436634179, 8.209049188425833992393494817631, 8.953296996381279759180169101143, 10.14897046210217559386370811700, 10.78382299096308958727270550056

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