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Properties

Label 2-384-12.11-c3-0-4
Degree $2$
Conductor $384$
Sign $0.608 - 0.793i$
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  + (−4.12 − 3.16i)3-s − 21.4i·5-s + 20.9i·7-s + (6.98 + 26.0i)9-s − 9.94·11-s − 67.8·13-s + (−67.7 + 88.2i)15-s − 7.97i·17-s + 62.4i·19-s + (66.1 − 86.1i)21-s + 101.·23-s − 333.·25-s + (53.7 − 129. i)27-s + 122. i·29-s − 87.5i·31-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)3-s − 1.91i·5-s + 1.12i·7-s + (0.258 + 0.965i)9-s − 0.272·11-s − 1.44·13-s + (−1.16 + 1.51i)15-s − 0.113i·17-s + 0.753i·19-s + (0.687 − 0.895i)21-s + 0.923·23-s − 2.66·25-s + (0.383 − 0.923i)27-s + 0.784i·29-s − 0.506i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6138277047\)
\(L(\frac12)\) \(\approx\) \(0.6138277047\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.12 + 3.16i)T \)
good5 \( 1 + 21.4iT - 125T^{2} \)
7 \( 1 - 20.9iT - 343T^{2} \)
11 \( 1 + 9.94T + 1.33e3T^{2} \)
13 \( 1 + 67.8T + 2.19e3T^{2} \)
17 \( 1 + 7.97iT - 4.91e3T^{2} \)
19 \( 1 - 62.4iT - 6.85e3T^{2} \)
23 \( 1 - 101.T + 1.21e4T^{2} \)
29 \( 1 - 122. iT - 2.43e4T^{2} \)
31 \( 1 + 87.5iT - 2.97e4T^{2} \)
37 \( 1 - 106.T + 5.06e4T^{2} \)
41 \( 1 - 90.3iT - 6.89e4T^{2} \)
43 \( 1 - 451. iT - 7.95e4T^{2} \)
47 \( 1 - 428.T + 1.03e5T^{2} \)
53 \( 1 + 362. iT - 1.48e5T^{2} \)
59 \( 1 + 801.T + 2.05e5T^{2} \)
61 \( 1 - 647.T + 2.26e5T^{2} \)
67 \( 1 - 957. iT - 3.00e5T^{2} \)
71 \( 1 + 224.T + 3.57e5T^{2} \)
73 \( 1 + 108.T + 3.89e5T^{2} \)
79 \( 1 - 615. iT - 4.93e5T^{2} \)
83 \( 1 + 204.T + 5.71e5T^{2} \)
89 \( 1 + 454. iT - 7.04e5T^{2} \)
97 \( 1 - 740.T + 9.12e5T^{2} \)
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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

−11.45884525756253604202837675151, −9.985862398825303994959024144149, −9.149544490454180391874691357576, −8.275472177948340336513966344416, −7.41320070624385627525527305202, −5.93394862395808217231034113989, −5.23307103957609244430581086726, −4.60178117160375283160184002128, −2.34625578077692356962765718291, −1.10116597853352790531279970653, 0.26322410670498656458733986950, 2.60650318823816314331371984823, 3.70641435692043354632351436083, 4.80543641335962468986212154299, 6.10216833403173679350668880583, 7.13610085350296271857668288621, 7.38070173214418943695908733700, 9.353384252485487239510879063883, 10.31253189035200203177987323864, 10.59954114495455196877933443865

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