Properties

Degree $2$
Conductor $1386$
Sign $-0.932 + 0.359i$
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 + (0.346 + 0.199i)5-s + (−1.03 + 2.43i)7-s + 0.999·8-s + (−0.346 + 0.199i)10-s + (−2.70 + 1.91i)11-s + 0.164i·13-s + (−1.58 − 2.11i)14-s + (−0.5 + 0.866i)16-s + (0.906 + 1.57i)17-s + (5.41 + 3.12i)19-s − 0.399i·20-s + (−0.308 − 3.30i)22-s + (−3.76 − 2.17i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.154 + 0.0894i)5-s + (−0.392 + 0.919i)7-s + 0.353·8-s + (−0.109 + 0.0632i)10-s + (−0.815 + 0.578i)11-s + 0.0456i·13-s + (−0.424 − 0.565i)14-s + (−0.125 + 0.216i)16-s + (0.219 + 0.380i)17-s + (1.24 + 0.717i)19-s − 0.0894i·20-s + (−0.0656 − 0.704i)22-s + (−0.785 − 0.453i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.932 + 0.359i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.932 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4738710940\)
\(L(\frac12)\) \(\approx\) \(0.4738710940\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 - 2.43i)T \)
11 \( 1 + (2.70 - 1.91i)T \)
good5 \( 1 + (-0.346 - 0.199i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.164iT - 13T^{2} \)
17 \( 1 + (-0.906 - 1.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.41 - 3.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.76 + 2.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.12T + 29T^{2} \)
31 \( 1 + (0.141 + 0.245i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.40 + 4.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.23T + 41T^{2} \)
43 \( 1 - 2.07iT - 43T^{2} \)
47 \( 1 + (0.367 + 0.212i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.71 - 4.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.92 - 3.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.10 - 3.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0327 - 0.0567i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.43iT - 71T^{2} \)
73 \( 1 + (7.21 - 4.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.531 + 0.306i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.91T + 83T^{2} \)
89 \( 1 + (8.89 + 5.13i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−9.889188173646022527659054910150, −9.268741092746662149922673512851, −8.256119737945429607374747043155, −7.74787905685661425590278141387, −6.76683331296561057577473917291, −5.83676300576194686552261901313, −5.38546965995357320015095206621, −4.18638242111564396138872424950, −2.89866118824062001041495324952, −1.78910927064300440469014340658, 0.21296625449597242969488000439, 1.54884409924130399177976544285, 3.00035459092470203000082730104, 3.61111199739938079106778677713, 4.85155019657770103944139205787, 5.67234038931188777437722804105, 6.89138478392115332202752800983, 7.63648397582154726107823484697, 8.288323391325861110843684378162, 9.561248366608185478836514007739

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