Properties

Label 2-387-43.42-c4-0-41
Degree $2$
Conductor $387$
Sign $-0.836 + 0.548i$
Analytic cond. $40.0041$
Root an. cond. $6.32488$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.72i·2-s − 29.1·4-s + 1.48i·5-s + 13.7i·7-s + 88.6i·8-s + 9.96·10-s + 10.2·11-s + 98.4·13-s + 92.7·14-s + 128.·16-s + 286.·17-s + 367. i·19-s − 43.2i·20-s − 68.5i·22-s + 242.·23-s + ⋯
L(s)  = 1  − 1.68i·2-s − 1.82·4-s + 0.0592i·5-s + 0.281i·7-s + 1.38i·8-s + 0.0996·10-s + 0.0843·11-s + 0.582·13-s + 0.473·14-s + 0.503·16-s + 0.991·17-s + 1.01i·19-s − 0.108i·20-s − 0.141i·22-s + 0.457·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.836 + 0.548i$
Analytic conductor: \(40.0041\)
Root analytic conductor: \(6.32488\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :2),\ -0.836 + 0.548i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.821550210\)
\(L(\frac12)\) \(\approx\) \(1.821550210\)
\(L(3)\) 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 \( 1 + (1.54e3 - 1.01e3i)T \)
good2 \( 1 + 6.72iT - 16T^{2} \)
5 \( 1 - 1.48iT - 625T^{2} \)
7 \( 1 - 13.7iT - 2.40e3T^{2} \)
11 \( 1 - 10.2T + 1.46e4T^{2} \)
13 \( 1 - 98.4T + 2.85e4T^{2} \)
17 \( 1 - 286.T + 8.35e4T^{2} \)
19 \( 1 - 367. iT - 1.30e5T^{2} \)
23 \( 1 - 242.T + 2.79e5T^{2} \)
29 \( 1 + 1.14e3iT - 7.07e5T^{2} \)
31 \( 1 - 895.T + 9.23e5T^{2} \)
37 \( 1 + 2.29e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.69e3T + 2.82e6T^{2} \)
47 \( 1 + 743.T + 4.87e6T^{2} \)
53 \( 1 + 99.3T + 7.89e6T^{2} \)
59 \( 1 + 3.28e3T + 1.21e7T^{2} \)
61 \( 1 + 3.22e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.55e3T + 2.01e7T^{2} \)
71 \( 1 + 2.95e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.61e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.38e3T + 3.89e7T^{2} \)
83 \( 1 - 6.42e3T + 4.74e7T^{2} \)
89 \( 1 + 3.29e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.32e3T + 8.85e7T^{2} \)
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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

−10.42936527590268536735719422405, −9.737496910309070931506742787961, −8.823998799782689850218857598848, −7.87258849402143375695153279624, −6.32837966109396170936982794234, −5.10362070044459467404858853769, −3.90164744387938728028638440616, −3.03876795644483399811647391654, −1.82490054733683233381669070392, −0.69217930902843848953149460929, 0.985852064485153469616146880462, 3.22881335146405453979412200360, 4.65899140191940907377385720165, 5.38134355907145543144310960653, 6.58769103969682160634735627630, 7.08467918830017323492622303252, 8.242773803333756040968086190781, 8.798862272427462318907830551432, 9.875087748144982281029058095683, 10.94412566940568168355790460116

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