Properties

Label 2-384-8.5-c3-0-16
Degree $2$
Conductor $384$
Sign $1$
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  + 3i·3-s + 5.67i·5-s + 33.0·7-s − 9·9-s − 34.6i·11-s − 82.2i·13-s − 17.0·15-s + 97.8·17-s − 55.8i·19-s + 99.2i·21-s − 130.·23-s + 92.7·25-s − 27i·27-s + 147. i·29-s − 101.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.507i·5-s + 1.78·7-s − 0.333·9-s − 0.949i·11-s − 1.75i·13-s − 0.293·15-s + 1.39·17-s − 0.674i·19-s + 1.03i·21-s − 1.18·23-s + 0.742·25-s − 0.192i·27-s + 0.944i·29-s − 0.586·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.342845514\)
\(L(\frac12)\) \(\approx\) \(2.342845514\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
good5 \( 1 - 5.67iT - 125T^{2} \)
7 \( 1 - 33.0T + 343T^{2} \)
11 \( 1 + 34.6iT - 1.33e3T^{2} \)
13 \( 1 + 82.2iT - 2.19e3T^{2} \)
17 \( 1 - 97.8T + 4.91e3T^{2} \)
19 \( 1 + 55.8iT - 6.85e3T^{2} \)
23 \( 1 + 130.T + 1.21e4T^{2} \)
29 \( 1 - 147. iT - 2.43e4T^{2} \)
31 \( 1 + 101.T + 2.97e4T^{2} \)
37 \( 1 + 184. iT - 5.06e4T^{2} \)
41 \( 1 - 237.T + 6.89e4T^{2} \)
43 \( 1 + 199. iT - 7.95e4T^{2} \)
47 \( 1 - 334.T + 1.03e5T^{2} \)
53 \( 1 + 102. iT - 1.48e5T^{2} \)
59 \( 1 + 105. iT - 2.05e5T^{2} \)
61 \( 1 - 717. iT - 2.26e5T^{2} \)
67 \( 1 - 316. iT - 3.00e5T^{2} \)
71 \( 1 + 800.T + 3.57e5T^{2} \)
73 \( 1 - 301.T + 3.89e5T^{2} \)
79 \( 1 - 42.8T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 505.T + 9.12e5T^{2} \)
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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

−10.68105211999744189783558151794, −10.45732231449068863559435062360, −8.946960106670096525422693897338, −8.102979100467464860501742148540, −7.45686821276609728758520869908, −5.69409115542563174472255821506, −5.23827461800257500343042988104, −3.83260629071676326944981521178, −2.70273725952707378567414619786, −0.942284741161076249285924439161, 1.34427720714433291050040697241, 2.03917448723456304114599333653, 4.16211174296075620475750147891, 4.93023411770112291767569032272, 6.09584053176418991870089823380, 7.44070302337166028436326400359, 7.957049846229254429169169749343, 8.941918527584486058874165456033, 9.953122371476158641052485191612, 11.15442584366350690946693824957

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