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Properties

Label 2-1386-7.4-c1-0-5
Degree $2$
Conductor $1386$
Sign $-0.749 - 0.661i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.20 + 2.09i)5-s + (1.62 − 2.09i)7-s − 0.999·8-s + (−1.20 + 2.09i)10-s + (0.5 − 0.866i)11-s − 6.82·13-s + (2.62 + 0.358i)14-s + (−0.5 − 0.866i)16-s + (−3.91 + 6.77i)17-s + (2.41 + 4.18i)19-s − 2.41·20-s + 0.999·22-s + (2.62 + 4.54i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.539 + 0.935i)5-s + (0.612 − 0.790i)7-s − 0.353·8-s + (−0.381 + 0.661i)10-s + (0.150 − 0.261i)11-s − 1.89·13-s + (0.700 + 0.0958i)14-s + (−0.125 − 0.216i)16-s + (−0.949 + 1.64i)17-s + (0.553 + 0.959i)19-s − 0.539·20-s + 0.213·22-s + (0.546 + 0.946i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.793853276\)
\(L(\frac12)\) \(\approx\) \(1.793853276\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.09i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.20 - 2.09i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 6.82T + 13T^{2} \)
17 \( 1 + (3.91 - 6.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.41 - 4.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.62 - 4.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (5.24 - 9.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.828 - 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 - 2.82T + 43T^{2} \)
47 \( 1 + (0.378 + 0.655i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.41 - 9.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.82 + 6.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.20 - 3.82i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.74 - 3.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 + (0.414 - 0.717i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.62 + 11.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.17T + 83T^{2} \)
89 \( 1 + (2.24 + 3.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−10.00317132171155064411588538507, −9.025638923056664694952054534158, −7.996850512832606669151519144619, −7.31423455827977500751907899775, −6.72154936986498187605117110998, −5.81244741522650276625272603062, −4.91916754104161788336795775564, −4.00041023137910432940469268598, −2.95874300717292081382872984800, −1.72812060966195884419057826533, 0.62620629884395417407779320090, 2.24456830159863625944498970796, 2.61357252844115598472692173870, 4.49925639639893120650247363208, 4.92873320033882545165258287102, 5.49057891625365421758971751259, 6.80476004567841863493852147232, 7.64995701679328947565621727535, 8.852374529401760582606701458086, 9.382659765370747703995964782898

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