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Properties

Label 2-1110-185.159-c1-0-15
Degree $2$
Conductor $1110$
Sign $-0.513 - 0.857i$
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 + (1.50 + 1.65i)5-s + 0.999i·6-s + (−0.827 − 0.478i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.682 + 2.12i)10-s + 0.252·11-s + (−0.866 + 0.499i)12-s + (0.858 − 1.48i)13-s − 0.956i·14-s + (0.473 + 2.18i)15-s + (−0.5 − 0.866i)16-s + (2.88 + 4.99i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.671 + 0.740i)5-s + 0.408i·6-s + (−0.312 − 0.180i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.215 + 0.673i)10-s + 0.0762·11-s + (−0.249 + 0.144i)12-s + (0.238 − 0.412i)13-s − 0.255i·14-s + (0.122 + 0.564i)15-s + (−0.125 − 0.216i)16-s + (0.698 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.513 - 0.857i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.513 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.411432623\)
\(L(\frac12)\) \(\approx\) \(2.411432623\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 1.65i)T \)
37 \( 1 + (3.20 + 5.16i)T \)
good7 \( 1 + (0.827 + 0.478i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.252T + 11T^{2} \)
13 \( 1 + (-0.858 + 1.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.88 - 4.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.89 - 2.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.71T + 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 + 0.279iT - 31T^{2} \)
41 \( 1 + (2.96 - 5.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.27T + 43T^{2} \)
47 \( 1 + 6.17iT - 47T^{2} \)
53 \( 1 + (7.10 - 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.0 + 7.52i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.43 - 1.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.518 + 0.299i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.17 + 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 + (2.20 + 1.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.5 - 7.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.84 + 3.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−10.11435094115459822044723717634, −9.322767107773799745459775019178, −8.329236317381566892373735986372, −7.62422529335408132603851404743, −6.70972664719274048563417909349, −5.91424696628807618449539698726, −5.16919032889327396831723200376, −3.70294503229561719446029698323, −3.28288410820823961998598460481, −1.82739458578989805809751399574, 0.936645442137941568413577194737, 2.14079410941128626254829994569, 3.08420477776121819531128979972, 4.23421799036176822596412625400, 5.21182371444366908851784326235, 6.00443101390937148137847332876, 7.01859590816614931870308329418, 8.100855785573109049372650885670, 8.948206493078990860470911012617, 9.708950380858226369533515885675

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