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Properties

Label 2-1110-185.159-c1-0-19
Degree $2$
Conductor $1110$
Sign $0.0305 + 0.999i$
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 + (1.83 + 1.28i)5-s + 0.999i·6-s + (−3.31 − 1.91i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (0.196 − 2.22i)10-s + 1.67·11-s + (0.866 − 0.499i)12-s + (1.56 − 2.70i)13-s + 3.82i·14-s + (−0.943 − 2.02i)15-s + (−0.5 − 0.866i)16-s + (2.75 + 4.78i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.818 + 0.574i)5-s + 0.408i·6-s + (−1.25 − 0.722i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.0622 − 0.704i)10-s + 0.505·11-s + (0.249 − 0.144i)12-s + (0.432 − 0.749i)13-s + 1.02i·14-s + (−0.243 − 0.523i)15-s + (−0.125 − 0.216i)16-s + (0.669 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0305 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0305 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0305 + 0.999i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.0305 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.071565819\)
\(L(\frac12)\) \(\approx\) \(1.071565819\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 - 1.28i)T \)
37 \( 1 + (1.06 - 5.98i)T \)
good7 \( 1 + (3.31 + 1.91i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 + (-1.56 + 2.70i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.75 - 4.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.27 + 0.737i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.21T + 23T^{2} \)
29 \( 1 + 2.97iT - 29T^{2} \)
31 \( 1 + 5.65iT - 31T^{2} \)
41 \( 1 + (-4.89 + 8.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 + (-2.49 + 1.44i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.20 + 3.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.55 + 3.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.933 - 0.539i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.807 + 1.39i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + (1.38 + 0.796i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.79 + 2.19i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.36 - 4.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−9.914384140321169073594748223844, −9.106719157319636956035820662603, −7.969227885370889714196274498683, −7.05814828081797850542734591178, −6.24750127637064026665248154649, −5.68416311015237762999088441754, −4.04494662105179991962197161401, −3.29332241480702939516451052185, −2.06365206570027762750675995924, −0.68941841414823323499711563240, 1.11855895720703199040885521618, 2.71118546888481468900615060226, 4.12080227901542686470321815700, 5.18638632917573845208590983173, 5.98970134159782658954947187443, 6.42641572225598171279768400265, 7.40021057884982036345804130575, 8.788170876626049585298235819526, 9.250420443666061854304100845325, 9.700354579244243847228755201834

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