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Properties

Label 2-1110-185.84-c1-0-32
Degree $2$
Conductor $1110$
Sign $0.873 - 0.487i$
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 + (1.40 + 1.74i)5-s + 0.999·6-s + (3.36 − 1.94i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.342 + 2.20i)10-s − 0.297·11-s + (0.866 + 0.499i)12-s + (−0.273 + 0.157i)13-s + 3.88·14-s + (2.08 + 0.808i)15-s + (−0.5 + 0.866i)16-s + (−0.766 − 0.442i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.626 + 0.779i)5-s + 0.408·6-s + (1.27 − 0.734i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.108 + 0.698i)10-s − 0.0896·11-s + (0.249 + 0.144i)12-s + (−0.0758 + 0.0438i)13-s + 1.03·14-s + (0.538 + 0.208i)15-s + (−0.125 + 0.216i)16-s + (−0.185 − 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.873 - 0.487i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.873 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.411921477\)
\(L(\frac12)\) \(\approx\) \(3.411921477\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 1.74i)T \)
37 \( 1 + (1.88 - 5.78i)T \)
good7 \( 1 + (-3.36 + 1.94i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.297T + 11T^{2} \)
13 \( 1 + (0.273 - 0.157i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.766 + 0.442i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.55 - 2.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.33iT - 23T^{2} \)
29 \( 1 + 4.88T + 29T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
41 \( 1 + (6.17 + 10.7i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 + 5.14iT - 47T^{2} \)
53 \( 1 + (-4.71 - 2.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.37 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.26 + 2.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.512 + 0.295i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.61 + 4.53i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 + (-1.22 - 2.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.61 - 3.24i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.65 + 4.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.87iT - 97T^{2} \)
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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

−10.09896783245786095629615062611, −8.872580187726968407468515833510, −8.060966256584084069670008011658, −7.32276901500634919133882894462, −6.68630251746464476723893317676, −5.67047169187590540492538772703, −4.70832685196075783026041320771, −3.75757188890237651060395918138, −2.61688048999397952772630616340, −1.60746295244707064070727762378, 1.52302575999547418527706163386, 2.28057391209532152827331932516, 3.56028866976577905162215860637, 4.78407696538296000798553064036, 5.17845599839440292969743766410, 6.03439093397903764420362760725, 7.43371152327785816309522113600, 8.262818647608944021143419042246, 9.104132630227254555969031091464, 9.616240449505994302574242638396

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