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Properties

Label 2-387-43.42-c4-0-8
Degree $2$
Conductor $387$
Sign $0.731 + 0.681i$
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  − 7.49i·2-s − 40.1·4-s + 9.40i·5-s − 53.5i·7-s + 180. i·8-s + 70.4·10-s − 151.·11-s − 319.·13-s − 401.·14-s + 713.·16-s − 32.2·17-s − 304. i·19-s − 377. i·20-s + 1.13e3i·22-s + 199.·23-s + ⋯
L(s)  = 1  − 1.87i·2-s − 2.50·4-s + 0.376i·5-s − 1.09i·7-s + 2.82i·8-s + 0.704·10-s − 1.24·11-s − 1.89·13-s − 2.04·14-s + 2.78·16-s − 0.111·17-s − 0.842i·19-s − 0.943i·20-s + 2.33i·22-s + 0.376·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.731 + 0.681i$
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: \((2,\ 387,\ (\ :2),\ 0.731 + 0.681i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5197838310\)
\(L(\frac12)\) \(\approx\) \(0.5197838310\)
\(L(3)\) 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 \( 1 + (-1.35e3 - 1.26e3i)T \)
good2 \( 1 + 7.49iT - 16T^{2} \)
5 \( 1 - 9.40iT - 625T^{2} \)
7 \( 1 + 53.5iT - 2.40e3T^{2} \)
11 \( 1 + 151.T + 1.46e4T^{2} \)
13 \( 1 + 319.T + 2.85e4T^{2} \)
17 \( 1 + 32.2T + 8.35e4T^{2} \)
19 \( 1 + 304. iT - 1.30e5T^{2} \)
23 \( 1 - 199.T + 2.79e5T^{2} \)
29 \( 1 - 268. iT - 7.07e5T^{2} \)
31 \( 1 - 665.T + 9.23e5T^{2} \)
37 \( 1 - 1.17e3iT - 1.87e6T^{2} \)
41 \( 1 + 212.T + 2.82e6T^{2} \)
47 \( 1 - 3.10e3T + 4.87e6T^{2} \)
53 \( 1 + 2.66e3T + 7.89e6T^{2} \)
59 \( 1 + 2.74e3T + 1.21e7T^{2} \)
61 \( 1 + 5.88e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.09e3T + 2.01e7T^{2} \)
71 \( 1 - 5.84e3iT - 2.54e7T^{2} \)
73 \( 1 - 663. iT - 2.83e7T^{2} \)
79 \( 1 - 6.32e3T + 3.89e7T^{2} \)
83 \( 1 + 8.35e3T + 4.74e7T^{2} \)
89 \( 1 - 4.25e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.58e3T + 8.85e7T^{2} \)
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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

−10.61981790473947899765256595320, −10.07317641953705075187233335631, −9.264891437935756963136005523551, −7.975233034242889214232581668035, −7.04868470422718398473727795293, −5.04799195065739526530706743090, −4.49135427805505362649017900470, −3.05420767650675940875295581927, −2.42998077974691466028076408877, −0.834167670446944956224127264603, 0.20183426346292785146182295993, 2.60673009032133549999537775833, 4.52468734624008621930906653803, 5.26698787357276762459897727587, 5.89859124207318632154953250230, 7.19196756524093836433965273841, 7.82381021083472719277502995910, 8.729857986830303969394499441147, 9.445883231461659026793473922431, 10.40790608786374206763170498428

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