Properties

Label 2-1110-185.64-c1-0-16
Degree $2$
Conductor $1110$
Sign $0.916 - 0.399i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (2.18 − 0.473i)5-s − 0.999i·6-s + (0.827 − 0.478i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.682 + 2.12i)10-s + 0.252·11-s + (0.866 + 0.499i)12-s + (−0.858 − 1.48i)13-s + 0.956i·14-s + (−1.65 + 1.50i)15-s + (−0.5 + 0.866i)16-s + (−2.88 + 4.99i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.977 − 0.211i)5-s − 0.408i·6-s + (0.312 − 0.180i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.215 + 0.673i)10-s + 0.0762·11-s + (0.249 + 0.144i)12-s + (−0.238 − 0.412i)13-s + 0.255i·14-s + (−0.427 + 0.387i)15-s + (−0.125 + 0.216i)16-s + (−0.698 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.381189405\)
\(L(\frac12)\) \(\approx\) \(1.381189405\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-2.18 + 0.473i)T \)
37 \( 1 + (-3.20 + 5.16i)T \)
good7 \( 1 + (-0.827 + 0.478i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.252T + 11T^{2} \)
13 \( 1 + (0.858 + 1.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.88 - 4.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.89 + 2.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.71T + 23T^{2} \)
29 \( 1 + 6.74iT - 29T^{2} \)
31 \( 1 - 0.279iT - 31T^{2} \)
41 \( 1 + (2.96 + 5.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 + 6.17iT - 47T^{2} \)
53 \( 1 + (-7.10 - 4.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-13.0 - 7.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.43 + 1.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.518 + 0.299i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.17 - 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 + (2.20 - 1.27i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.5 - 7.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.84 - 3.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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

−9.838436360432215572413650375815, −9.108925780582193971949043093967, −8.358767168794576167702197878859, −7.28759786595031153039256754836, −6.49292018353419446632917601674, −5.64243800611358480449544380825, −5.04319776647245003970970791287, −3.98470402903641291531416719269, −2.32360892794864643242114436079, −0.923726576746801729858935999304, 1.13445261338672183410223482396, 2.21303638511191670093688992821, 3.22196527941200832874124037412, 4.81315060430499051420036000453, 5.33586786639330150357256616837, 6.60972648351765833569310335467, 7.14323309237755843310320605636, 8.301623000871946210298043238640, 9.244341363858801269266428308678, 9.755688141709094576079377836582

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