Properties

Label 2-1386-33.2-c1-0-4
Degree $2$
Conductor $1386$
Sign $0.851 - 0.524i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.411 − 0.133i)5-s + (−0.587 + 0.809i)7-s + (−0.809 + 0.587i)8-s − 0.432i·10-s + (−0.137 + 3.31i)11-s + (−1.34 − 0.436i)13-s + (0.587 + 0.809i)14-s + (0.309 + 0.951i)16-s + (−0.223 − 0.688i)17-s + (3.54 + 4.87i)19-s + (−0.411 − 0.133i)20-s + (3.10 + 1.15i)22-s − 1.67i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.184 − 0.0598i)5-s + (−0.222 + 0.305i)7-s + (−0.286 + 0.207i)8-s − 0.136i·10-s + (−0.0414 + 0.999i)11-s + (−0.372 − 0.121i)13-s + (0.157 + 0.216i)14-s + (0.0772 + 0.237i)16-s + (−0.0542 − 0.166i)17-s + (0.812 + 1.11i)19-s + (−0.0920 − 0.0299i)20-s + (0.662 + 0.246i)22-s − 0.348i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434662408\)
\(L(\frac12)\) \(\approx\) \(1.434662408\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.137 - 3.31i)T \)
good5 \( 1 + (-0.411 + 0.133i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.34 + 0.436i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.223 + 0.688i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.54 - 4.87i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.67iT - 23T^{2} \)
29 \( 1 + (-0.367 - 0.266i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.99 - 6.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.85 - 4.98i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.35 + 0.985i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.52iT - 43T^{2} \)
47 \( 1 + (-2.88 - 3.97i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.253 - 0.0824i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.56 + 4.90i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.756 + 0.245i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.680T + 67T^{2} \)
71 \( 1 + (-2.38 + 0.774i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.78 - 5.20i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.90 + 1.92i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.83 - 5.64i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.71iT - 89T^{2} \)
97 \( 1 + (5.31 - 16.3i)T + (-78.4 - 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.791330030644352208935142729849, −9.132584429377116173136857887066, −8.060966002566163565235431041163, −7.27495906765478201306419152604, −6.20006684803280172129835852295, −5.33854379223683208506178704001, −4.53367912207464485839468717943, −3.47725928726663252937391136317, −2.47641999976447423557400020133, −1.40215664965545731434670667458, 0.56668747520225823793363118909, 2.44966824567379008561493413196, 3.55897713131361561276436292657, 4.45660675887177901147345310889, 5.55091446694037598344359540592, 6.10339548440748177655144626238, 7.12587361039027207654851516304, 7.70487496826425419170026994927, 8.673013958045139813516236899578, 9.387782771487058990564178096360

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