Properties

Degree $2$
Conductor $1386$
Sign $0.771 - 0.636i$
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 + (−1.42 + 0.822i)5-s + (−2.58 − 0.551i)7-s + 0.999·8-s + (1.42 + 0.822i)10-s + (0.578 − 3.26i)11-s + 1.29i·13-s + (0.816 + 2.51i)14-s + (−0.5 − 0.866i)16-s + (1.68 − 2.91i)17-s + (0.340 − 0.196i)19-s − 1.64i·20-s + (−3.11 + 1.13i)22-s + (−0.455 + 0.262i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.636 + 0.367i)5-s + (−0.978 − 0.208i)7-s + 0.353·8-s + (0.450 + 0.259i)10-s + (0.174 − 0.984i)11-s + 0.358i·13-s + (0.218 + 0.672i)14-s + (−0.125 − 0.216i)16-s + (0.408 − 0.707i)17-s + (0.0780 − 0.0450i)19-s − 0.367i·20-s + (−0.664 + 0.241i)22-s + (−0.0949 + 0.0548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.771 - 0.636i$
Motivic weight: \(1\)
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7127516058\)
\(L(\frac12)\) \(\approx\) \(0.7127516058\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 + 0.551i)T \)
11 \( 1 + (-0.578 + 3.26i)T \)
good5 \( 1 + (1.42 - 0.822i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.29iT - 13T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.340 + 0.196i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.455 - 0.262i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.33T + 29T^{2} \)
31 \( 1 + (2.86 - 4.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.50 - 2.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + (4.48 - 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.43 + 0.830i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.26 - 4.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.32 + 1.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.62iT - 71T^{2} \)
73 \( 1 + (-8.82 - 5.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.22 - 4.17i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.98T + 83T^{2} \)
89 \( 1 + (8.32 - 4.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−9.611378923679825991217965298837, −9.084503912637001804640338417694, −8.090849756174066902084922689722, −7.33371478260385956350831585003, −6.53698841867875453797424789041, −5.53637692694169678572129068476, −4.21634449497930444032945886165, −3.41002476490684133153722571961, −2.74537282949296879767988719954, −1.01078977118601783639992572103, 0.40849674188887091447336875581, 2.11839831716821857064120233080, 3.60255542303819139829497549327, 4.36419271796939132853024541725, 5.51689429361667521769962012047, 6.23388976124624923462077545952, 7.19190482516353411306078114954, 7.79624929740816447429092890194, 8.639660585103364600084975273380, 9.460611026994348268787676875273

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