Properties

Label 2-387-43.17-c1-0-11
Degree $2$
Conductor $387$
Sign $-0.723 + 0.690i$
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  + (−0.377 − 1.65i)2-s + (−0.789 + 0.380i)4-s + (−0.140 + 0.358i)5-s + (1.74 − 3.02i)7-s + (−1.18 − 1.48i)8-s + (0.646 + 0.0974i)10-s + (3.90 + 1.88i)11-s + (1.26 − 0.190i)13-s + (−5.65 − 1.74i)14-s + (−3.10 + 3.89i)16-s + (−0.205 − 0.523i)17-s + (−6.30 − 4.29i)19-s + (−0.0252 − 0.336i)20-s + (1.63 − 7.16i)22-s + (−0.553 − 7.39i)23-s + ⋯
L(s)  = 1  + (−0.266 − 1.16i)2-s + (−0.394 + 0.190i)4-s + (−0.0629 + 0.160i)5-s + (0.659 − 1.14i)7-s + (−0.420 − 0.526i)8-s + (0.204 + 0.0308i)10-s + (1.17 + 0.567i)11-s + (0.350 − 0.0528i)13-s + (−1.51 − 0.466i)14-s + (−0.776 + 0.974i)16-s + (−0.0498 − 0.127i)17-s + (−1.44 − 0.986i)19-s + (−0.00564 − 0.0753i)20-s + (0.348 − 1.52i)22-s + (−0.115 − 1.54i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.723 + 0.690i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ -0.723 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474988 - 1.18590i\)
\(L(\frac12)\) \(\approx\) \(0.474988 - 1.18590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 5.66i)T \)
good2 \( 1 + (0.377 + 1.65i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.140 - 0.358i)T + (-3.66 - 3.40i)T^{2} \)
7 \( 1 + (-1.74 + 3.02i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.90 - 1.88i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.26 + 0.190i)T + (12.4 - 3.83i)T^{2} \)
17 \( 1 + (0.205 + 0.523i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (6.30 + 4.29i)T + (6.94 + 17.6i)T^{2} \)
23 \( 1 + (0.553 + 7.39i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (3.26 + 1.00i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (0.717 - 0.665i)T + (2.31 - 30.9i)T^{2} \)
37 \( 1 + (2.19 + 3.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.07 - 4.72i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (3.93 - 1.89i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-9.56 - 1.44i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (-2.92 + 3.66i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-3.96 - 3.67i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-10.8 - 7.39i)T + (24.4 + 62.3i)T^{2} \)
71 \( 1 + (0.570 - 7.61i)T + (-70.2 - 10.5i)T^{2} \)
73 \( 1 + (2.05 - 0.309i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-4.14 + 7.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.1 + 3.43i)T + (68.5 - 46.7i)T^{2} \)
89 \( 1 + (2.60 - 0.804i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + (-0.441 - 0.212i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.95523798186950410941668267435, −10.40471212204688388382960785221, −9.344746223266921408178493024164, −8.518523714096959770008700568083, −7.10611669916927662414492441896, −6.46532282414001578374698162216, −4.53870024630830329884198398537, −3.83926541018216205092394152442, −2.31250493941865304490343709504, −1.00181551000083311528972962223, 1.99282819620771614343629257939, 3.78969581033177148717694437617, 5.30850574501986892386189287294, 6.01467140489722052173295973252, 6.87564872377489497458583892904, 8.161987217279936934695606151977, 8.611674553982328811509196044702, 9.353960206928095575379056315257, 10.87333263823669572590922033001, 11.75514955405247633845080232918

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