Properties

Label 2-1110-185.64-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.554 - 0.831i$
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 + (0.603 + 2.15i)5-s + 0.999i·6-s + (0.998 − 0.576i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−2.16 − 0.553i)10-s + 1.60·11-s + (−0.866 − 0.499i)12-s + (1.89 + 3.28i)13-s + 1.15i·14-s + (1.59 + 1.56i)15-s + (−0.5 + 0.866i)16-s + (3.28 − 5.69i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.270 + 0.962i)5-s + 0.408i·6-s + (0.377 − 0.217i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.685 − 0.175i)10-s + 0.482·11-s + (−0.249 − 0.144i)12-s + (0.526 + 0.911i)13-s + 0.308i·14-s + (0.412 + 0.403i)15-s + (−0.125 + 0.216i)16-s + (0.797 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.554 - 0.831i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.554 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.842291542\)
\(L(\frac12)\) \(\approx\) \(1.842291542\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.603 - 2.15i)T \)
37 \( 1 + (-6.07 - 0.267i)T \)
good7 \( 1 + (-0.998 + 0.576i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.60T + 11T^{2} \)
13 \( 1 + (-1.89 - 3.28i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.28 + 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.174 - 0.100i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.633T + 23T^{2} \)
29 \( 1 + 5.73iT - 29T^{2} \)
31 \( 1 - 8.34iT - 31T^{2} \)
41 \( 1 + (-4.00 - 6.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.95T + 43T^{2} \)
47 \( 1 + 4.65iT - 47T^{2} \)
53 \( 1 + (7.47 + 4.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.3 - 5.99i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.33 + 5.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.52 + 9.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.32iT - 73T^{2} \)
79 \( 1 + (-6.91 + 3.99i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.4 - 7.21i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.6 - 9.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.93T + 97T^{2} \)
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

−9.632210493609342026220662849637, −9.281108732517653621941080462381, −8.131296317766317640492726538047, −7.49660873680814496765274323972, −6.71284528538434101916815371937, −6.11531729072944784288100688323, −4.86318960291973891578559802859, −3.73843455418590700637931625477, −2.59472459038736861665307397459, −1.29930228658222603458791576779, 1.07651545216126841568642361630, 2.11410062117615854874065074957, 3.47466738653128929050811286067, 4.24108590727469478415084590452, 5.33456668197039811557478533239, 6.15644192333128685714143970413, 7.80596169254367792924262298957, 8.169000028126776573773228124714, 9.003758376453661995100869536372, 9.601022492071137704382712623046

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