Properties

Label 2-1110-185.84-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.154 - 0.988i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.363 + 2.20i)5-s + 0.999·6-s + (−0.268 + 0.154i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.41 + 1.72i)10-s + 4.57·11-s + (0.866 + 0.499i)12-s + (−3.32 + 1.91i)13-s − 0.309·14-s + (0.788 + 2.09i)15-s + (−0.5 + 0.866i)16-s + (2.86 + 1.65i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.162 + 0.986i)5-s + 0.408·6-s + (−0.101 + 0.0585i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.448 + 0.546i)10-s + 1.37·11-s + (0.249 + 0.144i)12-s + (−0.921 + 0.531i)13-s − 0.0827·14-s + (0.203 + 0.540i)15-s + (−0.125 + 0.216i)16-s + (0.695 + 0.401i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.154 - 0.988i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.154 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.675821878\)
\(L(\frac12)\) \(\approx\) \(2.675821878\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.363 - 2.20i)T \)
37 \( 1 + (-6.02 - 0.841i)T \)
good7 \( 1 + (0.268 - 0.154i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.57T + 11T^{2} \)
13 \( 1 + (3.32 - 1.91i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.86 - 1.65i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.688 + 1.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.47iT - 23T^{2} \)
29 \( 1 + 0.690T + 29T^{2} \)
31 \( 1 - 2.73T + 31T^{2} \)
41 \( 1 + (-0.175 - 0.303i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.04iT - 43T^{2} \)
47 \( 1 - 3.77iT - 47T^{2} \)
53 \( 1 + (11.8 + 6.84i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.77 + 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.37 - 4.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.13 + 2.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.67 + 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 + (-7.63 - 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.66 - 1.53i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.47 + 9.48i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.990iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.768711722291545339516768739021, −9.350115458801346876047557198830, −8.055265178167595738548019395727, −7.46537987666938803619374832968, −6.60053863745705004359337128040, −6.12629414239461845880086043381, −4.72046185365047454741017032735, −3.74022074650142729830085348633, −2.99747486924289552050703249324, −1.78012587502798277123978827998, 0.971800970266334830175940904546, 2.34677235236564180915173248591, 3.55035881027290393319799840180, 4.35034469765501423719274682718, 5.08992279552439702353974082495, 6.09801409133684299286082575805, 7.17309205350868967198590225627, 8.128559652109132246293761128025, 8.960755633838122593291028972494, 9.694436957101812555793924479831

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