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Properties

Label 2-1110-185.159-c1-0-11
Degree $2$
Conductor $1110$
Sign $0.906 - 0.422i$
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 + (−2.16 + 0.575i)5-s − 0.999i·6-s + (−1.29 − 0.746i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−1.57 − 1.58i)10-s − 4.99·11-s + (0.866 − 0.499i)12-s + (3.55 − 6.16i)13-s − 1.49i·14-s + (2.15 + 0.581i)15-s + (−0.5 − 0.866i)16-s + (2.24 + 3.88i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.966 + 0.257i)5-s − 0.408i·6-s + (−0.488 − 0.282i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.499 − 0.500i)10-s − 1.50·11-s + (0.249 − 0.144i)12-s + (0.987 − 1.70i)13-s − 0.399i·14-s + (0.557 + 0.150i)15-s + (−0.125 − 0.216i)16-s + (0.544 + 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.906 - 0.422i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.101007613\)
\(L(\frac12)\) \(\approx\) \(1.101007613\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 - 0.575i)T \)
37 \( 1 + (-4.87 - 3.64i)T \)
good7 \( 1 + (1.29 + 0.746i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + (-3.55 + 6.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.24 - 3.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.68 - 1.55i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 - 8.24iT - 29T^{2} \)
31 \( 1 + 0.925iT - 31T^{2} \)
41 \( 1 + (-3.23 + 5.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 8.04T + 43T^{2} \)
47 \( 1 + 4.51iT - 47T^{2} \)
53 \( 1 + (-3.66 + 2.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.55 - 2.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.38 - 4.83i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.5 - 6.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.259 + 0.448i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + (0.836 + 0.482i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.83 + 4.52i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.79 + 3.92i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.48T + 97T^{2} \)
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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

−10.31237904143602741638304223763, −8.675243908656265485468867498077, −7.992553530728477402536128617635, −7.44599867005794767398596245667, −6.59355476499264768846784315501, −5.55020404172263386516383048256, −5.04936391123911043139455589454, −3.56165425786780214427405877254, −3.07787579634467752835353336043, −0.72507937397459379640176675874, 0.821562796179361951458184484692, 2.63999476128362572113422567577, 3.56792987595577529613187797314, 4.60448392729224923669240574535, 5.18852825493596378130987497628, 6.30371910610784745134814633699, 7.23925936029805279847650773367, 8.215601665015733302615436533705, 9.264839478183697763598471263967, 9.723241256935066560688298774819

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