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Properties

Label 2-1110-185.159-c1-0-22
Degree $2$
Conductor $1110$
Sign $0.185 - 0.982i$
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 + (0.653 + 2.13i)5-s − 0.999i·6-s + (3.20 + 1.84i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−1.52 + 1.63i)10-s + 4.44·11-s + (0.866 − 0.499i)12-s + (3.09 − 5.36i)13-s + 3.69i·14-s + (0.503 − 2.17i)15-s + (−0.5 − 0.866i)16-s + (−1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.292 + 0.956i)5-s − 0.408i·6-s + (1.20 + 0.698i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.482 + 0.517i)10-s + 1.33·11-s + (0.249 − 0.144i)12-s + (0.858 − 1.48i)13-s + 0.987i·14-s + (0.129 − 0.562i)15-s + (−0.125 − 0.216i)16-s + (−0.405 − 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.185 - 0.982i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.152807117\)
\(L(\frac12)\) \(\approx\) \(2.152807117\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.653 - 2.13i)T \)
37 \( 1 + (-1.62 - 5.86i)T \)
good7 \( 1 + (-3.20 - 1.84i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.44T + 11T^{2} \)
13 \( 1 + (-3.09 + 5.36i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.88 - 2.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.31T + 23T^{2} \)
29 \( 1 + 0.108iT - 29T^{2} \)
31 \( 1 + 3.58iT - 31T^{2} \)
41 \( 1 + (-0.667 + 1.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.49T + 43T^{2} \)
47 \( 1 - 1.22iT - 47T^{2} \)
53 \( 1 + (7.70 - 4.44i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.292 - 0.169i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.7 + 7.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.66 + 4.61i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.67iT - 73T^{2} \)
79 \( 1 + (4.87 + 2.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.20 + 3.00i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.9 - 8.03i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−10.04981556521661622862985604925, −9.091761335514895355765698930971, −8.062914561960793906949232565399, −7.55100451748575540591205047523, −6.46492030170059787485728842136, −5.87011944261587744554465381167, −5.18487121008944855790467735021, −3.96324635490546268907876770287, −2.83960438138805394500891849916, −1.42015018352187914903114604587, 1.16215630131932845263850474417, 1.76006749567064363622139957321, 3.91333190171930869708620337533, 4.26653895966145712554163063107, 5.10788435084454126470511578325, 6.10261480170367683016061894896, 6.97496544446009610058910305980, 8.284301300820834579798679056709, 9.055964958580770868790037707663, 9.614316709545126032068606635902

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