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Properties

Label 2-384-3.2-c2-0-8
Degree $2$
Conductor $384$
Sign $-0.296 - 0.955i$
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  + (2.86 − 0.888i)3-s + 8.59i·5-s − 10.9·7-s + (7.41 − 5.09i)9-s + 2.75i·11-s + 4.43·13-s + (7.63 + 24.6i)15-s + 25.4i·17-s − 17.5·19-s + (−31.3 + 9.71i)21-s + 17.5i·23-s − 48.8·25-s + (16.7 − 21.1i)27-s + 19.6i·29-s − 2.58·31-s + ⋯
L(s)  = 1  + (0.955 − 0.296i)3-s + 1.71i·5-s − 1.56·7-s + (0.824 − 0.565i)9-s + 0.250i·11-s + 0.341·13-s + (0.509 + 1.64i)15-s + 1.49i·17-s − 0.923·19-s + (−1.49 + 0.462i)21-s + 0.762i·23-s − 1.95·25-s + (0.619 − 0.784i)27-s + 0.676i·29-s − 0.0833·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.296 - 0.955i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.296 - 0.955i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.955652 + 1.29702i\)
\(L(\frac12)\) \(\approx\) \(0.955652 + 1.29702i\)
\(L(2)\) 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 \( 1 + (-2.86 + 0.888i)T \)
good5 \( 1 - 8.59iT - 25T^{2} \)
7 \( 1 + 10.9T + 49T^{2} \)
11 \( 1 - 2.75iT - 121T^{2} \)
13 \( 1 - 4.43T + 169T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 + 17.5T + 361T^{2} \)
23 \( 1 - 17.5iT - 529T^{2} \)
29 \( 1 - 19.6iT - 841T^{2} \)
31 \( 1 + 2.58T + 961T^{2} \)
37 \( 1 + 7.73T + 1.36e3T^{2} \)
41 \( 1 - 58.0iT - 1.68e3T^{2} \)
43 \( 1 - 42.1T + 1.84e3T^{2} \)
47 \( 1 + 17.4iT - 2.20e3T^{2} \)
53 \( 1 + 69.0iT - 2.80e3T^{2} \)
59 \( 1 + 50.5iT - 3.48e3T^{2} \)
61 \( 1 - 32.5T + 3.72e3T^{2} \)
67 \( 1 - 48.0T + 4.48e3T^{2} \)
71 \( 1 - 22.1iT - 5.04e3T^{2} \)
73 \( 1 + 27.0T + 5.32e3T^{2} \)
79 \( 1 - 97.4T + 6.24e3T^{2} \)
83 \( 1 - 59.5iT - 6.88e3T^{2} \)
89 \( 1 + 110. iT - 7.92e3T^{2} \)
97 \( 1 - 55.1T + 9.40e3T^{2} \)
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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

−11.20189576167226399931483890714, −10.25790530271654413993368048488, −9.764522146748426907842650275322, −8.619603331299865495267745479538, −7.51643794377982699056779978318, −6.61886610278226755632490782827, −6.20902010552454160318232866480, −3.79040558303223486085321330655, −3.26488069102010351935595793005, −2.13288167011758992012355743752, 0.61284391498372858234643930832, 2.48586382215703987785121047212, 3.79539159312775555207300583753, 4.71210281140747890797516423120, 5.95084133607178430978167312000, 7.22628575291274436621275482472, 8.397472600062487076133982541785, 9.108833762655520983519066781487, 9.533066332434789754066467712278, 10.58074826123443640296310867559

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