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Properties

Label 2-1110-185.64-c1-0-22
Degree $2$
Conductor $1110$
Sign $0.787 + 0.616i$
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 + (−2.12 − 0.690i)5-s + 0.999i·6-s + (3.89 − 2.24i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.66 − 1.49i)10-s + 0.412·11-s + (−0.866 − 0.499i)12-s + (1.10 + 1.90i)13-s + 4.49i·14-s + (−2.18 + 0.465i)15-s + (−0.5 + 0.866i)16-s + (−1.63 + 2.82i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.951 − 0.308i)5-s + 0.408i·6-s + (1.47 − 0.849i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.525 − 0.473i)10-s + 0.124·11-s + (−0.249 − 0.144i)12-s + (0.305 + 0.529i)13-s + 1.20i·14-s + (−0.564 + 0.120i)15-s + (−0.125 + 0.216i)16-s + (−0.395 + 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.511065918\)
\(L(\frac12)\) \(\approx\) \(1.511065918\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 0.690i)T \)
37 \( 1 + (-5.42 + 2.74i)T \)
good7 \( 1 + (-3.89 + 2.24i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.412T + 11T^{2} \)
13 \( 1 + (-1.10 - 1.90i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.63 - 2.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 1.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 + 9.16iT - 29T^{2} \)
31 \( 1 + 6.51iT - 31T^{2} \)
41 \( 1 + (1.68 + 2.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.32T + 43T^{2} \)
47 \( 1 + 6.71iT - 47T^{2} \)
53 \( 1 + (-10.0 - 5.82i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.76 - 2.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.46 + 3.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.71 + 1.56i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.01 + 1.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.57iT - 73T^{2} \)
79 \( 1 + (7.72 - 4.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.38 - 1.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.49 + 1.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−9.551951459350400229078725913331, −8.505058055067676750481348554950, −8.145143413723527124166130263695, −7.47624596192486853993305420251, −6.75263292094668905231219243624, −5.49307191369980488576775324056, −4.30064088283493885117453829258, −3.98425916440393649132862786975, −2.03381642320731032023800809090, −0.809664401547555983614031397905, 1.39824831185356547008083687326, 2.67099386804785233465914626707, 3.51169985260463785228083508240, 4.63874415838376780919775642404, 5.29369257254109334175936029244, 6.90580980288080762239355619979, 7.80817591864912110637085996804, 8.443852199176112661743772630907, 8.867944133746822745586877824077, 10.00817142155787881502766708121

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