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Properties

Label 2-387-43.38-c1-0-14
Degree $2$
Conductor $387$
Sign $-0.297 + 0.954i$
Analytic cond. $3.09021$
Root an. cond. $1.75789$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.178 − 0.780i)2-s + (1.22 + 0.589i)4-s + (−0.511 − 1.30i)5-s + (−2.37 − 4.12i)7-s + (1.67 − 2.10i)8-s + (−1.10 + 0.167i)10-s + (−2.39 + 1.15i)11-s + (0.611 + 0.0920i)13-s + (−3.64 + 1.12i)14-s + (0.352 + 0.441i)16-s + (2.15 − 5.50i)17-s + (−1.48 + 1.01i)19-s + (0.142 − 1.89i)20-s + (0.474 + 2.07i)22-s + (−0.178 + 2.37i)23-s + ⋯
L(s)  = 1  + (0.125 − 0.551i)2-s + (0.612 + 0.294i)4-s + (−0.228 − 0.583i)5-s + (−0.899 − 1.55i)7-s + (0.592 − 0.743i)8-s + (−0.350 + 0.0528i)10-s + (−0.722 + 0.348i)11-s + (0.169 + 0.0255i)13-s + (−0.972 + 0.300i)14-s + (0.0880 + 0.110i)16-s + (0.523 − 1.33i)17-s + (−0.340 + 0.231i)19-s + (0.0318 − 0.424i)20-s + (0.101 + 0.442i)22-s + (−0.0371 + 0.495i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.297 + 0.954i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ -0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849737 - 1.15497i\)
\(L(\frac12)\) \(\approx\) \(0.849737 - 1.15497i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.54 + 0.452i)T \)
good2 \( 1 + (-0.178 + 0.780i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.511 + 1.30i)T + (-3.66 + 3.40i)T^{2} \)
7 \( 1 + (2.37 + 4.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.39 - 1.15i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.611 - 0.0920i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (-2.15 + 5.50i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (1.48 - 1.01i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (0.178 - 2.37i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (-1.79 + 0.555i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-5.49 - 5.09i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (-1.11 + 1.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.583 + 2.55i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-8.07 - 3.88i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (13.5 - 2.04i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-4.23 - 5.30i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-3.03 + 2.81i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-4.65 + 3.17i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (-0.641 - 8.55i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (-8.41 - 1.26i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-3.09 - 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.21 - 0.374i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (-9.04 - 2.79i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (-4.06 + 1.95i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.96865483551758509567585747825, −10.28347915855055378287140290649, −9.577552574871249040755002923092, −8.071920581159700458250789763883, −7.28446120198340546074015551318, −6.53568262293835506954778206427, −4.86816905762086466913419547338, −3.83036320365127886281842699956, −2.81367412948000693736233523096, −0.929570601979941830699958093929, 2.29149649336743260258213243520, 3.24011734715935209766228135464, 5.12752104089204155913096021339, 6.13085423968459849090005487160, 6.51144755062410785434540472522, 7.87882641675853354891180512282, 8.609427338150773461548176802819, 9.890529230647695426677684379075, 10.69438278576178273072576363894, 11.56513618797020243238561067365

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