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Properties

Label 2-1386-7.4-c1-0-9
Degree $2$
Conductor $1386$
Sign $-0.820 - 0.572i$
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 + (1.37 + 2.37i)5-s + (−1.37 + 2.26i)7-s − 0.999·8-s + (−1.37 + 2.37i)10-s + (0.5 − 0.866i)11-s + 5.49·13-s + (−2.64 − 0.0585i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (4.01 + 6.96i)19-s − 2.74·20-s + 0.999·22-s + (−0.645 − 1.11i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.614 + 1.06i)5-s + (−0.519 + 0.854i)7-s − 0.353·8-s + (−0.434 + 0.752i)10-s + (0.150 − 0.261i)11-s + 1.52·13-s + (−0.706 − 0.0156i)14-s + (−0.125 − 0.216i)16-s + (0.121 − 0.210i)17-s + (0.921 + 1.59i)19-s − 0.614·20-s + 0.213·22-s + (−0.134 − 0.232i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.820 - 0.572i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.820 - 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.058532104\)
\(L(\frac12)\) \(\approx\) \(2.058532104\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 - 2.26i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.37 - 2.37i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.49T + 13T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.01 - 6.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.645 + 1.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.54T + 29T^{2} \)
31 \( 1 + (2.54 - 4.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.01 - 3.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.54T + 41T^{2} \)
43 \( 1 + 4.03T + 43T^{2} \)
47 \( 1 + (4.64 + 8.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.74 + 4.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.76 - 8.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.829 - 1.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.77 - 3.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.54T + 71T^{2} \)
73 \( 1 + (-4.29 + 7.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.11 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + (3.25 + 5.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0872T + 97T^{2} \)
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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

−9.918072700194274068655393528622, −8.962571927186156893540630160626, −8.303653197388050544965812772354, −7.28329325476683865915002168051, −6.34078796921556825565133797213, −6.01881266574770801683925561300, −5.21694306981016880856537558860, −3.58672772644776853918932395368, −3.21015468608521083762286225699, −1.79920726663430106721304094073, 0.78269220352060580761730316496, 1.70527046813312249582710211818, 3.20083201620226236588913291561, 4.04199420513766855944301247050, 4.94061394688219252873275905449, 5.77556296752932755866938475045, 6.62116726250057689452119837104, 7.66208817623988685097412047665, 8.769147961279720835789673623543, 9.384484096267960207605753746444

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