Properties

Degree $2$
Conductor $1386$
Sign $-0.386 + 0.922i$
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 + (−1.5 − 2.59i)5-s + (2.5 + 0.866i)7-s + 0.999·8-s + (−1.5 + 2.59i)10-s + (−0.5 + 0.866i)11-s + 2·13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (−1 − 1.73i)19-s + 3·20-s + 0.999·22-s + (1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s + (0.944 + 0.327i)7-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (−0.150 + 0.261i)11-s + 0.554·13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.229 − 0.397i)19-s + 0.670·20-s + 0.213·22-s + (0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.386 + 0.922i$
Motivic weight: \(1\)
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257285605\)
\(L(\frac12)\) \(\approx\) \(1.257285605\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-8 + 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−9.106582593319319140327175679174, −8.646188225349877601386588116476, −7.960369256014594624864973877121, −7.24094646244902765700639984972, −5.78715347848264763768852222171, −4.79876517842271712229745900127, −4.34670305938235612015213356893, −3.07480895586425640386497719543, −1.76966996698982048694031002079, −0.68392592750658741446396216996, 1.25211510885093977073377401536, 2.81638326254037990007750585255, 3.89553828636340115483740286384, 4.78046277483799343820175957142, 5.94864695055717207207822762676, 6.67322426898016020530935443680, 7.47037256122227930271297546017, 8.150354029420569066497710908951, 8.659725080321463544619447621246, 9.949354119307875869852054270915

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