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Properties

Label 2-387-43.22-c0-0-0
Degree $2$
Conductor $387$
Sign $0.996 - 0.0789i$
Analytic cond. $0.193138$
Root an. cond. $0.439474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 + 0.781i)4-s − 1.94i·7-s + (0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s + (−0.678 + 0.541i)19-s + (−0.900 + 0.433i)25-s + (1.52 − 1.21i)28-s + (−1.62 − 0.781i)31-s − 1.56i·37-s + (0.900 − 0.433i)43-s − 2.80·49-s + (−0.777 + 0.974i)52-s + (0.678 + 1.40i)61-s + (−0.900 + 0.433i)64-s + (0.277 + 0.347i)67-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)4-s − 1.94i·7-s + (0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s + (−0.678 + 0.541i)19-s + (−0.900 + 0.433i)25-s + (1.52 − 1.21i)28-s + (−1.62 − 0.781i)31-s − 1.56i·37-s + (0.900 − 0.433i)43-s − 2.80·49-s + (−0.777 + 0.974i)52-s + (0.678 + 1.40i)61-s + (−0.900 + 0.433i)64-s + (0.277 + 0.347i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.996 - 0.0789i$
Analytic conductor: \(0.193138\)
Root analytic conductor: \(0.439474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :0),\ 0.996 - 0.0789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9179419879\)
\(L(\frac12)\) \(\approx\) \(0.9179419879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + 1.94iT - T^{2} \)
11 \( 1 + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.900 - 0.433i)T^{2} \)
19 \( 1 + (0.678 - 0.541i)T + (0.222 - 0.974i)T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
37 \( 1 + 1.56iT - T^{2} \)
41 \( 1 + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.900 - 0.433i)T^{2} \)
61 \( 1 + (-0.678 - 1.40i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
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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

−11.29707039052007006541196787845, −10.91535666635041557724485265297, −9.860607275660252158552658217962, −8.707972005462628394362828837401, −7.45720930067224604865458893520, −7.20965740068177585831527334287, −6.04501949723655223305687385693, −4.13422779959580015541245015690, −3.78822688227793451436260835727, −1.90305191097641647955838317118, 2.00546506124317938350757554901, 3.03713585885310433316305981298, 5.06195962311433660364737582025, 5.76766154129259029203947750022, 6.52092312491202000515826251929, 7.930821782551537661439080954167, 8.853966111725592023537413247111, 9.708213103319798961404065928638, 10.71265523763832106689210056989, 11.50159083721209636515446212428

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