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Properties

Label 2-387-43.37-c2-0-20
Degree $2$
Conductor $387$
Sign $-0.215 + 0.976i$
Analytic cond. $10.5449$
Root an. cond. $3.24730$
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  − 2.66i·2-s − 3.10·4-s + (2.37 + 1.36i)5-s + (2.06 − 1.19i)7-s − 2.39i·8-s + (3.64 − 6.31i)10-s + 18.5·11-s + (6.91 + 11.9i)13-s + (−3.17 − 5.49i)14-s − 18.7·16-s + (12.0 + 20.8i)17-s + (−7.60 − 4.39i)19-s + (−7.34 − 4.24i)20-s − 49.5i·22-s + (15.1 − 26.3i)23-s + ⋯
L(s)  = 1  − 1.33i·2-s − 0.775·4-s + (0.474 + 0.273i)5-s + (0.294 − 0.170i)7-s − 0.299i·8-s + (0.364 − 0.631i)10-s + 1.69·11-s + (0.532 + 0.921i)13-s + (−0.226 − 0.392i)14-s − 1.17·16-s + (0.706 + 1.22i)17-s + (−0.400 − 0.231i)19-s + (−0.367 − 0.212i)20-s − 2.25i·22-s + (0.660 − 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(10.5449\)
Root analytic conductor: \(3.24730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34478 - 1.67361i\)
\(L(\frac12)\) \(\approx\) \(1.34478 - 1.67361i\)
\(L(2)\) 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 \( 1 + (-16.4 - 39.7i)T \)
good2 \( 1 + 2.66iT - 4T^{2} \)
5 \( 1 + (-2.37 - 1.36i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.06 + 1.19i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 - 18.5T + 121T^{2} \)
13 \( 1 + (-6.91 - 11.9i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.0 - 20.8i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (7.60 + 4.39i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.1 + 26.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.40 + 5.43i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-17.1 + 29.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (57.1 + 33.0i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 61.5T + 1.68e3T^{2} \)
47 \( 1 - 60.3T + 2.20e3T^{2} \)
53 \( 1 + (-23.0 + 39.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 51.2T + 3.48e3T^{2} \)
61 \( 1 + (-78.4 + 45.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (43.4 - 75.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (14.1 - 8.19i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (24.4 - 14.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-9.04 - 15.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-8.69 + 15.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-67.8 - 39.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 32.8T + 9.40e3T^{2} \)
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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

−10.86160179461695617813197993355, −10.19573868730017924234005508031, −9.261198500583372244756719005994, −8.490517991370191259251354459548, −6.81255714001057083085053091069, −6.20670224376960395253563289421, −4.36971432746472060690109836298, −3.67697051421233298095501872065, −2.17713005291996716848211550794, −1.20132822756999069095399006091, 1.42845687552901143220697919162, 3.40635148271390212186419344815, 4.99622222142442094220915312698, 5.65634653606392692008625684732, 6.67993119511966261244309267411, 7.43255038700827132539524779982, 8.613716749539625437275919607683, 9.103042461035445969309215620265, 10.26791493129339920336942828009, 11.54160416773676693504337051956

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