Properties

Label 2-1110-185.174-c1-0-5
Degree $2$
Conductor $1110$
Sign $-0.781 - 0.623i$
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 + (1.64 + 1.51i)5-s + 0.999·6-s + (1.09 + 0.631i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.18 − 0.494i)10-s − 4.99·11-s + (−0.866 + 0.499i)12-s + (4.64 + 2.68i)13-s − 1.26·14-s + (−0.662 − 2.13i)15-s + (−0.5 − 0.866i)16-s + (−3.69 + 2.13i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.734 + 0.679i)5-s + 0.408·6-s + (0.413 + 0.238i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.689 − 0.156i)10-s − 1.50·11-s + (−0.249 + 0.144i)12-s + (1.28 + 0.743i)13-s − 0.337·14-s + (−0.171 − 0.551i)15-s + (−0.125 − 0.216i)16-s + (−0.895 + 0.516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.781 - 0.623i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7029083375\)
\(L(\frac12)\) \(\approx\) \(0.7029083375\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 1.51i)T \)
37 \( 1 + (-5.93 + 1.32i)T \)
good7 \( 1 + (-1.09 - 0.631i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + (-4.64 - 2.68i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.69 - 2.13i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.67 - 6.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.94iT - 23T^{2} \)
29 \( 1 - 0.263T + 29T^{2} \)
31 \( 1 + 8.35T + 31T^{2} \)
41 \( 1 + (-3.74 + 6.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 - 3.66iT - 47T^{2} \)
53 \( 1 + (7.06 - 4.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.45 + 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.390 + 0.675i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 - 1.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.141 - 0.244i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.54iT - 73T^{2} \)
79 \( 1 + (4.05 - 7.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.9 - 6.88i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.52 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.02iT - 97T^{2} \)
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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

−10.36560991343164038274100830641, −9.309389926847616240064504527423, −8.384914887777768768551025236577, −7.80950654938223024072918239032, −6.61727609404196841481744212672, −6.17207783087529199506753294798, −5.40996068090019756622005048064, −4.17059897026232960500756924552, −2.48808874343640172834337186499, −1.67035458287016544316508328208, 0.38842667956626580711845355556, 1.76832589161977493746644034543, 2.98215428024847378960959808473, 4.38544566166314310024937313408, 5.25438461911662957937555842596, 5.97521844327512785157529078699, 7.15217121782122453245800499339, 8.063985414760829139485016470081, 8.862103708284028487310862383353, 9.499693739107610529628318047347

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