↑

Properties

Label 2-1110-185.64-c1-0-25
Degree $2$
Conductor $1110$
Sign $0.211 + 0.977i$
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 + (1.64 + 1.51i)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 + (2.18 + 0.491i)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 + (0.734 + 0.678i)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 + (0.563 + 0.126i)15-s + (−0.125 + 0.216i)16-s + (0.179 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.211 + 0.977i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.211 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.621579254\)
\(L(\frac12)\) \(\approx\) \(2.621579254\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 1.51i)T \)
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

−9.577448913579996822822249877293, −9.263264582840767576255180752103, −7.78747834857698267118489942467, −7.37622049276347482620179240738, −6.13019948054742800737349990299, −5.38473803215621397313139690344, −4.31042282936499068002592919564, −2.95474496127804779919179234645, −2.55413178794276525548994028529, −1.10002704318452504280456660775, 1.57888109698743060906846464971, 2.83049692581539653794391222826, 4.07312165640815351305672130580, 4.99882002941586491847652941477, 5.55752585681621551460057830034, 6.66002009342672735996576202292, 7.58878591370927189480221552424, 8.417710793480719175883220692049, 9.124164833416734124927864788988, 9.743091123774481247861144941506

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