↑

Properties

Label 2-1110-185.64-c1-0-24
Degree $2$
Conductor $1110$
Sign $-0.380 + 0.924i$
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 + (−0.491 − 2.18i)5-s − 0.999i·6-s + (−0.916 + 0.529i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (2.13 + 0.664i)10-s − 0.825·11-s + (0.866 + 0.499i)12-s + (1.81 + 3.14i)13-s − 1.05i·14-s + (1.51 + 1.64i)15-s + (−0.5 + 0.866i)16-s + (−0.742 + 1.28i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.219 − 0.975i)5-s − 0.408i·6-s + (−0.346 + 0.200i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.675 + 0.210i)10-s − 0.248·11-s + (0.249 + 0.144i)12-s + (0.504 + 0.873i)13-s − 0.282i·14-s + (0.391 + 0.424i)15-s + (−0.125 + 0.216i)16-s + (−0.179 + 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.380 + 0.924i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.380 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3146546527\)
\(L(\frac12)\) \(\approx\) \(0.3146546527\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.491 + 2.18i)T \)
37 \( 1 + (5.95 - 1.24i)T \)
good7 \( 1 + (0.916 - 0.529i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.825T + 11T^{2} \)
13 \( 1 + (-1.81 - 3.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.742 - 1.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.75 + 2.74i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 + 2.02iT - 29T^{2} \)
31 \( 1 + 6.27iT - 31T^{2} \)
41 \( 1 + (1.42 + 2.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + (8.00 + 4.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.33 + 3.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.60 - 3.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.2 + 6.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.701 + 1.21i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.82iT - 73T^{2} \)
79 \( 1 + (3.10 - 1.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (13.4 + 7.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.86 + 1.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.18T + 97T^{2} \)
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

−9.511395291931700330689911615638, −8.778035545665449286518986674302, −8.053048542683021374793116743395, −7.03101849593370856563247403548, −6.18581945992860366225680930425, −5.33948872827656287177369982595, −4.59992454524907753605871579510, −3.58429485466107062207766322916, −1.69147855454952929946713738084, −0.17363405708796233750335647578, 1.46321153945933637419821022594, 2.96391697417696267698893358301, 3.53601329609954516396231721208, 4.93854700125065592615118695317, 5.97850687253864224195657656784, 6.86393245241146725078437759386, 7.64336607012408117798539339735, 8.350699870042752951730594052312, 9.588041254772642593988318160866, 10.27974996691340714455388051908

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