Properties

Label 2-387-43.13-c1-0-15
Degree $2$
Conductor $387$
Sign $-0.969 + 0.243i$
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  + (0.483 − 2.11i)2-s + (−2.45 − 1.18i)4-s + (−0.0260 − 0.00392i)5-s + (1.56 − 2.71i)7-s + (−0.984 + 1.23i)8-s + (−0.0208 + 0.0532i)10-s + (−3.36 + 1.62i)11-s + (−1.78 − 4.55i)13-s + (−4.98 − 4.62i)14-s + (−1.25 − 1.57i)16-s + (5.84 − 0.881i)17-s + (0.296 + 3.95i)19-s + (0.0592 + 0.0404i)20-s + (1.80 + 7.91i)22-s + (−0.284 − 0.193i)23-s + ⋯
L(s)  = 1  + (0.342 − 1.49i)2-s + (−1.22 − 0.591i)4-s + (−0.0116 − 0.00175i)5-s + (0.591 − 1.02i)7-s + (−0.348 + 0.436i)8-s + (−0.00660 + 0.0168i)10-s + (−1.01 + 0.488i)11-s + (−0.496 − 1.26i)13-s + (−1.33 − 1.23i)14-s + (−0.314 − 0.394i)16-s + (1.41 − 0.213i)17-s + (0.0680 + 0.908i)19-s + (0.0132 + 0.00903i)20-s + (0.385 + 1.68i)22-s + (−0.0592 − 0.0403i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184880 - 1.49782i\)
\(L(\frac12)\) \(\approx\) \(0.184880 - 1.49782i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.54 + 0.341i)T \)
good2 \( 1 + (-0.483 + 2.11i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.0260 + 0.00392i)T + (4.77 + 1.47i)T^{2} \)
7 \( 1 + (-1.56 + 2.71i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.36 - 1.62i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.78 + 4.55i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (-5.84 + 0.881i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.296 - 3.95i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (0.284 + 0.193i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (0.971 + 0.901i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-2.06 + 0.638i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (-5.01 - 8.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.431 + 1.89i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-10.0 - 4.82i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.12 + 5.41i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (0.107 + 0.134i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-5.31 - 1.63i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (0.0822 + 1.09i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (-4.37 + 2.98i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (2.70 + 6.88i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-2.70 + 4.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.49 + 1.38i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (2.93 - 2.72i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (8.11 - 3.90i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.78561299862151907055726616011, −10.22113185698640590670835975775, −9.755366118489198284350333061028, −7.918565688116857560285002449074, −7.56631304411341876802156050398, −5.62817768550234352544663141865, −4.66123389377964377665984216281, −3.61065178328422334284286292787, −2.48014053021725023799436730331, −0.957410967868940165128991704979, 2.38201679346544145626225786563, 4.21775294403563306546343865652, 5.39642302984874295303359986497, 5.77107535141704343442116861230, 7.11538328622989799884936530884, 7.83611420781773244290124216782, 8.682865337569999276170504225419, 9.537915710083733376470291657214, 10.95725571088163988176319859338, 11.84517598633305151288267732712

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