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Properties

Label 2-1110-185.159-c1-0-16
Degree $2$
Conductor $1110$
Sign $0.967 + 0.252i$
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 + (2.17 − 0.503i)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 + (2.13 + 0.653i)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 + (0.974 − 0.225i)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 + (0.552 + 0.168i)15-s + (−0.125 − 0.216i)16-s + (0.405 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.967 + 0.252i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774345183\)
\(L(\frac12)\) \(\approx\) \(1.774345183\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.503i)T \)
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

−9.691045099219746134200675091178, −9.367285421153761567911805655229, −8.569453110102827923878268033098, −7.18135671428696103216543392569, −6.69422219783861839461726663848, −5.51660931416518991793554077448, −4.16800808190336528135134763254, −3.59267638214787009837529466914, −2.34215159361771168568558746023, −1.23669090646041699943481260570, 1.02406423111633606888891510552, 2.66436692723937098449501307926, 3.28346541702877938094635223031, 5.10205736875354308664708254999, 5.74479636563358248772409446534, 6.77323878699263196584793830832, 7.09630903631340709748412771629, 8.341522123022823061905808355728, 9.220661207844677789465340781975, 9.637566185748953774264121818840

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