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Properties

Label 2-1386-21.17-c1-0-8
Degree $2$
Conductor $1386$
Sign $0.805 - 0.592i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.36 + 2.35i)5-s + (−2.13 − 1.56i)7-s + 0.999i·8-s + (−2.35 − 1.36i)10-s + (0.866 + 0.5i)11-s − 0.193i·13-s + (2.63 + 0.284i)14-s + (−0.5 − 0.866i)16-s + (3.07 − 5.32i)17-s + (0.269 − 0.155i)19-s + 2.72·20-s − 0.999·22-s + (4.50 − 2.60i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.608 + 1.05i)5-s + (−0.807 − 0.590i)7-s + 0.353i·8-s + (−0.745 − 0.430i)10-s + (0.261 + 0.150i)11-s − 0.0537i·13-s + (0.703 + 0.0759i)14-s + (−0.125 − 0.216i)16-s + (0.745 − 1.29i)17-s + (0.0617 − 0.0356i)19-s + 0.608·20-s − 0.213·22-s + (0.939 − 0.542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.309664029\)
\(L(\frac12)\) \(\approx\) \(1.309664029\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.13 + 1.56i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-1.36 - 2.35i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.193iT - 13T^{2} \)
17 \( 1 + (-3.07 + 5.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.269 + 0.155i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.50 + 2.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.17iT - 29T^{2} \)
31 \( 1 + (-1.38 - 0.797i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.21 - 3.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.29T + 41T^{2} \)
43 \( 1 - 4.10T + 43T^{2} \)
47 \( 1 + (-3.63 - 6.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.04 + 0.603i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.43 + 9.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.20 - 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.69 + 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.63iT - 71T^{2} \)
73 \( 1 + (2.50 + 1.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.60 + 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + (-5.67 - 9.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.77iT - 97T^{2} \)
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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

−9.555087804841152681906890084847, −9.160376898311957763328320621330, −7.85303796844478432862206563029, −7.08275481880209651017606764063, −6.63877908370090200640235706871, −5.81930777397182456070345343294, −4.70628521503893907878993826140, −3.27530040718647482114963912137, −2.58338860409822625688836593396, −0.943023741120741072784472513409, 0.922739676594969770741070572623, 2.04192304701122853091127326407, 3.22455042621397403300130330824, 4.26782341922726421759378179649, 5.57793026501678487156696974135, 6.01954431323728611945985551328, 7.16395291229414438864747003018, 8.181629546392057294184318397318, 8.896332316244986261741277229470, 9.411371207262005479360478864949

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