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Properties

Label 2-387-43.37-c2-0-29
Degree $2$
Conductor $387$
Sign $-0.900 + 0.433i$
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  − 1.61i·2-s + 1.39·4-s + (0.333 + 0.192i)5-s + (−8.76 + 5.06i)7-s − 8.70i·8-s + (0.310 − 0.538i)10-s + 7.33·11-s + (−10.6 − 18.3i)13-s + (8.16 + 14.1i)14-s − 8.44·16-s + (−12.4 − 21.5i)17-s + (−6.15 − 3.55i)19-s + (0.467 + 0.269i)20-s − 11.8i·22-s + (3.21 − 5.56i)23-s + ⋯
L(s)  = 1  − 0.806i·2-s + 0.349·4-s + (0.0667 + 0.0385i)5-s + (−1.25 + 0.723i)7-s − 1.08i·8-s + (0.0310 − 0.0538i)10-s + 0.666·11-s + (−0.815 − 1.41i)13-s + (0.583 + 1.01i)14-s − 0.527·16-s + (−0.732 − 1.26i)17-s + (−0.324 − 0.187i)19-s + (0.0233 + 0.0134i)20-s − 0.537i·22-s + (0.139 − 0.242i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.900 + 0.433i$
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: \((2,\ 387,\ (\ :1),\ -0.900 + 0.433i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.274399 - 1.20227i\)
\(L(\frac12)\) \(\approx\) \(0.274399 - 1.20227i\)
\(L(2)\) 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 \( 1 + (21.0 - 37.4i)T \)
good2 \( 1 + 1.61iT - 4T^{2} \)
5 \( 1 + (-0.333 - 0.192i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (8.76 - 5.06i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 - 7.33T + 121T^{2} \)
13 \( 1 + (10.6 + 18.3i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (12.4 + 21.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (6.15 + 3.55i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.21 + 5.56i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-46.5 + 26.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (1.72 - 2.99i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-8.83 - 5.10i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 53.2T + 1.68e3T^{2} \)
47 \( 1 + 59.5T + 2.20e3T^{2} \)
53 \( 1 + (42.5 - 73.7i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 74.1T + 3.48e3T^{2} \)
61 \( 1 + (-5.41 + 3.12i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-14.6 + 25.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-85.2 + 49.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (75.1 - 43.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-72.0 - 124. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-6.84 + 11.8i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-108. - 62.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 92.7T + 9.40e3T^{2} \)
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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

−10.72576366369949218993300093109, −9.792986721127886091626397861326, −9.361775303927537671306305854578, −7.953172403039392990153474116858, −6.69922144560582724258760893627, −6.11677013040484414252198325657, −4.57724102670752131821995287361, −3.03083892736843532907096183275, −2.54070116816252235397390770886, −0.50430840577493944978842391823, 1.90374254411316099699911550390, 3.51750218062628084382876886757, 4.70128088883397533963143618588, 6.28959048393303156292965510266, 6.60782564198387881684677220044, 7.46516398483708776955091316671, 8.705220606609620162731176606773, 9.564801982914501028704140882458, 10.54424140473866756399676999231, 11.48774266685418085735737811273

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