Properties

Degree $2$
Conductor $1386$
Sign $-0.446 - 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (3.19 + 1.84i)5-s + (1.52 + 2.15i)7-s + 0.999·8-s + (−3.19 + 1.84i)10-s + (2.64 + 2.00i)11-s + 5.60i·13-s + (−2.63 + 0.245i)14-s + (−0.5 + 0.866i)16-s + (−1.39 − 2.41i)17-s + (2.14 + 1.23i)19-s − 3.69i·20-s + (−3.05 + 1.28i)22-s + (−3.26 − 1.88i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.42 + 0.825i)5-s + (0.578 + 0.815i)7-s + 0.353·8-s + (−1.01 + 0.583i)10-s + (0.797 + 0.603i)11-s + 1.55i·13-s + (−0.704 + 0.0656i)14-s + (−0.125 + 0.216i)16-s + (−0.338 − 0.585i)17-s + (0.492 + 0.284i)19-s − 0.825i·20-s + (−0.651 + 0.274i)22-s + (−0.680 − 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.446 - 0.894i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.926935796\)
\(L(\frac12)\) \(\approx\) \(1.926935796\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.52 - 2.15i)T \)
11 \( 1 + (-2.64 - 2.00i)T \)
good5 \( 1 + (-3.19 - 1.84i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.60iT - 13T^{2} \)
17 \( 1 + (1.39 + 2.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.14 - 1.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.26 + 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.51T + 29T^{2} \)
31 \( 1 + (1.70 + 2.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.85 + 4.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 7.72iT - 43T^{2} \)
47 \( 1 + (8.98 + 5.18i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.27 + 5.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.6 + 6.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.66 - 9.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.15iT - 71T^{2} \)
73 \( 1 + (-4.53 + 2.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.250 - 0.144i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (13.7 + 7.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.58T + 97T^{2} \)
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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

−9.622702199068930642626139940371, −9.181341827570570932296186387735, −8.368120612434773818070495509467, −7.02882611116030933934880003233, −6.74147288076020380451050510910, −5.81880168184209378788508170770, −5.12623331354641789896900448174, −3.96061315532844428860650116131, −2.24612777222618454801015388163, −1.81264517176520757521958969440, 0.952285538011062467770473567130, 1.65054749986256317120245237839, 2.99766389672233853016436068219, 4.11094129728605886953865685606, 5.15372106026147072062442282893, 5.82627216702321176243095374420, 6.86955204455146519105424931186, 8.124680263841655675193171476614, 8.486243516328221534651310286920, 9.567634653973579434770334726943

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