Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Sign $-0.820 + 0.572i$
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 + (1.37 − 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

\( d \)  =  \(2\)
\( N \)  =  \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-0.820 + 0.572i$
motivic weight  =  \(1\)
character  :  $\chi_{1386} (793, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1386,\ (\ :1/2),\ -0.820 + 0.572i)\)
\(L(1)\)  \(\approx\)  \(2.058532104\)
\(L(\frac12)\)  \(\approx\)  \(2.058532104\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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