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Properties

Label 2-1110-185.174-c1-0-35
Degree $2$
Conductor $1110$
Sign $-0.937 + 0.346i$
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 + (−1.86 + 1.23i)5-s + 0.999·6-s + (−3.46 − 2i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1 + 2i)10-s + 11-s + (0.866 − 0.499i)12-s + (−2.59 − 1.5i)13-s − 3.99·14-s + (−2.23 + 0.133i)15-s + (−0.5 − 0.866i)16-s + (−5.19 + 3i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.834 + 0.550i)5-s + 0.408·6-s + (−1.30 − 0.755i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.316 + 0.632i)10-s + 0.301·11-s + (0.249 − 0.144i)12-s + (−0.720 − 0.416i)13-s − 1.06·14-s + (−0.576 + 0.0345i)15-s + (−0.125 − 0.216i)16-s + (−1.26 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.937 + 0.346i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.937 + 0.346i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6942445943\)
\(L(\frac12)\) \(\approx\) \(0.6942445943\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 - 1.23i)T \)
37 \( 1 + (6.06 + 0.5i)T \)
good7 \( 1 + (3.46 + 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (2.59 + 1.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.19 - 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + (1.73 - i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.66 + 5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.5 - 4.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−9.586552162231000667925736613849, −8.834884481179102684965816455307, −7.60135758710877619022448860490, −6.93382898625983724569592332420, −6.26051792534973769158794296497, −4.78226975012131178775640686162, −3.98052925093805615692513134377, −3.29496070802326476320960186252, −2.41043263288142054287824514091, −0.21459013233408683789015183469, 2.03914371950139317304233224052, 3.30090896050323077553340695764, 3.88904228834475255816826712674, 5.11389822303637143725518061486, 5.94752062365395676645696621462, 7.10201365484174287963941947498, 7.40288247634945459900036376093, 8.651670714849529385558616158693, 9.186250955605158470569214696365, 9.889753548707353559031523740361

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