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Properties

Label 2-1110-185.64-c1-0-26
Degree $2$
Conductor $1110$
Sign $-0.0999 + 0.994i$
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  + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (2.23 + 0.0669i)5-s + 0.999i·6-s + (3.29 − 1.90i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.17 − 1.90i)10-s − 3.84·11-s + (0.866 + 0.499i)12-s + (−1.22 − 2.12i)13-s − 3.80i·14-s + (−1.96 + 1.05i)15-s + (−0.5 + 0.866i)16-s + (3.42 − 5.93i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.999 + 0.0299i)5-s + 0.408i·6-s + (1.24 − 0.718i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.371 − 0.601i)10-s − 1.16·11-s + (0.249 + 0.144i)12-s + (−0.339 − 0.588i)13-s − 1.01i·14-s + (−0.508 + 0.273i)15-s + (−0.125 + 0.216i)16-s + (0.831 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0999 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0999 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.0999 + 0.994i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.0999 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.967190637\)
\(L(\frac12)\) \(\approx\) \(1.967190637\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0669i)T \)
37 \( 1 + (4.13 + 4.46i)T \)
good7 \( 1 + (-3.29 + 1.90i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.84T + 11T^{2} \)
13 \( 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.42 + 5.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.996 - 0.575i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.0656T + 23T^{2} \)
29 \( 1 + 2.70iT - 29T^{2} \)
31 \( 1 - 8.12iT - 31T^{2} \)
41 \( 1 + (-0.239 - 0.415i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 1.94iT - 47T^{2} \)
53 \( 1 + (4.83 + 2.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.87 - 1.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.07 - 2.35i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.70 + 3.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.72 + 6.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.82iT - 73T^{2} \)
79 \( 1 + (-2.57 + 1.48i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.83 + 2.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.89 - 3.98i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.90T + 97T^{2} \)
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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

−9.970909194453790869992808873246, −9.072859868557450896553481040815, −7.904577762583667210688222685189, −7.16164163896221708779615851327, −5.83718219353754019733462492093, −5.10549210233200393879949001396, −4.72439669488406655319021435293, −3.23852936423211678636854239478, −2.15612511379406354300552705961, −0.858301905581995102830360009148, 1.64357840854953715897699402905, 2.58575394058712101384188945951, 4.31561603589746724989141026291, 5.26788547691434256385324250541, 5.66183284174451877824347979493, 6.49331925175439159143796126995, 7.63475915097151915102803750569, 8.207987979953554245716793143849, 9.083682023001775355001920113894, 10.12342607681736570649361561507

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